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TT&5C TsH {English Translation with Mathematical Explanations & Notes) vol. II Present second volume translates all the verses in English. Translation is not literal but in mathematical terms, but preserving the technical terms in Sanskrit. Verses in praise of god have been left out, not because of disrespect. With all devotion inspired by Samanta Chandrashekhar, this is not the purpose of the second volume. In addition to trans- lation, each formula has been explained or derived accord- ing to modern mathematics and astronomy. The methods have been compared with other Indian astronomers and some times with other countries and with modern astronomy. This was the method and purpose, of Samanta himself. Technical terms and their calculations cannot be explained in words alone. So a general mathematical and technical introduction is given at beginning of each chapter with bibliography or source reference for further study. In that light only, the methods proved in the chap- ter can be understood. Where-ever considered use- ful, methods have also been explained with examples, based on text as well as modern astronomy. [ ' «l'ldlRuQ ^C^ft^T^FT ^°fp)^ ^ 3te snftrfr 3T«ifa fas!?* |^K fW^3R(ftsfcH i|*i)^dmf*TrTT3 I TCHT ft*c|ftai?U 3t qijjM <MHlfd %^5R0T *j)fd+ ft*H 3T«^R^f 3TCT, 3Tcf: W* 3 *rTCcffa ^ ^ tfaR M^fl^l $ ^^Rf^SHId^TlT tj%T 3rffcf I3?*ft 3^3fM<^lf4d ^wt fw aik "5? ?ft ^^r§t $ f^reft, te& srgjf f^r ^ ^r ^5 m #-^ SIDDHANTA-DARPANA (1899 A.D.) English Translation with Mathematical Explainations and Notes Vol. II Arun Kumar Upadhyay, IPS M.Sc, AIFC NAG PUBLISHERS 11A/U.A., Jawahar Nagar, Delhi-1 10007 This publication has been published with the financial assistance by Rashtriya Sanskrit Sansthan, New Delhi. Nag Publishers (i) 11A/U.A. (Post Offio Nagar, Delhi 110007. (ii) Sanskrit Bhawan, 12, 15, Sanskrit Nagar Plot No. 3, Sector-14, Rohini, New Delhi-85 (Hi) Jalalpur Mafi, Chunar, Dist. Mirzapur, U.P. © Aran Kumar Upadhyay, IPS B-9, CB-9, Cantonment Road, Cuttack-753001 ISBN 81-7081-342-9 (Set) ISBN 81-7081-406-1 (Vol II) Price : Rs. First Edition : 1998 PRINTED IN INDIA Published by Shri Surendra Pratap for Nag Publishers, X^T^T^^S^ Bagh, Dayabasti, Delhi - 110035 SUBJECT INDEX Chapter Verses No. Subject Pages No. INTRODUCTION 1-25 1. Arrangement of book 1 2. Numeration 2 3. Transliteration of Sanskrta letters 5 4. Survey of Indian Astronomy 6 5. Astronomers of Orissa ' 12 6. Candrasekhara 17 7. References 24 A. MADHYAMADHIKARA 1. MEASUREMENT OF TIME 26-45 1A Introduction 1. Units of Measurement 26 2. Units of length 27 3. Measurement of time 31 4. Accurate measurements 34 5. Angles 36 6. References 38 1. TEXT 1-22 Importance of jyotisa etc. 38 23-39 Units of time 40 40-56 Current time, angle units 43 Units of length and time in 7 pages w Siddhanta Darpana 2. REVOLUTION OF PLANETS 46-58 2A. Introduction 45 1-26 Translation of Text 53 3. MEAN PLANETS 59.77 1-13 Ahargana 59 14-20 Lords of days etc. 62 21-23 Calculation of Mean position 64 24-46 Guru varsa 66 47-51 Other methods 71 52-70 Dhruva positions 73 71-78 Hara, end 76 4. CALCULATION AT DIFFERENT 78-112 PLACES 4A. Introduction and references 78 1-21 Sphuta paridhi, palabha, desantara 88 22-25 Cara correction 97 26-32 Bhujantara correction 101 33-41 Udayantara 105 42-58 Padaka, dhruva, end 110 B. SPHUTADHIKARA 5. TRUE PLANETS 113-249 5A. Introduction 1. Concepts of planetary motion 113 2. Modern Calculations 115 3. Tables of sun, planets 137 4. Equation for other planets 153 5. References 155 TEXT 1-42 Nature of planetary motion, types 156 43-75 Sara and its calculation 167 Subject Index v 76-89 Parocca of mangala, budha, sani 189 90-112 Sighra and manda paridhi 195 113-120 Bhuja, Koti phala 207 121-123 True position of star planets 209 124-131 Special methods for mangala, budha 214 132-138 True sun, moon speeds 217 139-158 True speed of tara graha 219 159-160 Vakri position 225 161-165 Heliacal setting 229 166 Mean from true planet 232 167-188 Use of Tables 232 189-193 Udaya and asta times 236 194-206 Calendar elements 238 207-212 Extra and lost months etc. - 246 6. CORRECTIONS TO MOON 250-379 6A. Introduction 1. Elliptical orbit of moon 250 2. Deviations due to sun 252 3. Corrections in defferent texts 254 4. Modern charts of moon 259 5. Indian charts 260 6. Making a calendar 261 7. Solar calendars in history 271 8. Luni-solar calendars 276 9. Old Indian calendars 281 10. Indian eras 293 11. Festivals and yogas 305 12. References • 318 TEXT 1-16 Extra 3 corrections to moon 319 336 348 Siddhanta Darpana 17-26 Accurate motion of moon 322 27-46 Need for accuracy, lambana 326 47- 57 Origin from Bhaskara, smrtis 329 58-67 Unequal naksatras 332 68-74 Sankranti, different orbits 335 75-91 Ayanamsa 92-101 Kranti 102-112 Variation in day length 352 113-117 Udayantara calculation 35g 118-120 Rising time of planets 359 121-130 Equator rising of rasis 360 131-142 Rising at other places 367 143-151 Dasama lagna 372 152-156 Rising of nirayana rasis in Orissa 374 157-162 Charts and end 376 C. TRIPRASNADHIKARA THREE PROBLEMS OF 380-475 DAILY MOTION 1-5 Cardinal directions 6. Sanku and chaya 7-10 Square root and multiplication 395 11-13 Setting of sanku, correct chaya 402 14-23 Definitions 24-27 Kranti from palabha 28-34 Sun from shadow and vice versa 410 35-38 Unmandala sanku 412 39-44 Agra, karna vrttagra 41 7 45-51 Sama mandala 42j 52-66 Kona sanku, natamsa 425 67-80 Shadow form time & vice versa 430 384 394 404 408 Subject Index v a 81-84 Sun form agra & samasanku 437 85 Path of shadow end 438 86-87 Lapsed part of night 440 88-92 Rising times in Orissa 441 93-95 Conclusion and end 442 7B. APPENDIX 1. True, mean and standard time 443 2. Equation of time 447 3. Parallax 450 4. Refraction 461 8. LUNAR ECLIPSE 476-545 TEXT 1-6 Possibility of eclipse , 476 7-11 Correct samaparva Kala 497 12-15 Mean diameter etc. of sun, moon 500 16-24 True diameter & distances 505 25-27 Earth shadow 510 28-33 Rahu and eclipse 513 34-38 Moon sara, its gati 515 39-40 Grasa, direction of eclipse 519 41-43 Duration of eclipse 521 44-45 Single step calculation 526 46-54 Grasa at different times 528 55-59 Direction of eclipse 531 60-65 Lunar day length 536 66-69 Valana correction 538 70-77 Diagram 539 78-83 True earth shadow 541 84-88 Colour of eclipse 544 Vlll 9. Siddhanta Darpana 10. 11 Solar Eclipse 1-6 Lambana and nati 7-15 - Sphuta lambana 16-22 Single step method 23-39 Nati correction in sara 40-45 Tamomana, true moon bimba 46-47 Hara 48-49 Eclipse at different places 50-57 True sthiti kala 58-60 Single step method for duration 61-62 Annular eclipse 63-64 Reason for extra methods 65-72 Eclipse duration through yasti 73-82 Misc corrections 83-85 Modern methods 86-89 Duration limits and end PARILEKHA 1-3 Valana 4-5 Angular measure of bimba 6-14 Diagram of eclipse 15-30 Grasa calculation 30-38 Another diagram, end CONJUNCTION OF PLANETS 1-9 Meaning and methods 10-11 Sara and pata 12-26 Correction to sara 27-31 Ayana drkkarma 31-37 Aksa drkkarma » 32-42 Diameter and bimba 43-55 Mean and observed bimba 546-599 546 547 555 558 565 570 572 573 576 576 577 577 581 583 592 600-612 600 602 604 607 610 613-651 613 618 622 624 626 630 635 Subject Index IX 56-60 Types of conjunction 640 61-67 Nati 641 68-71 Lambana * 643 72-90 Conjunction with moon, stars 643 91-96 Parilekha 646 97-108 Observing planet shadows 647 109 Increase of bright birhba 649 110-112 Solar eclipse by sukra, end 650 12. CONJUNCTIONS WITH STARS 652-681 TEXT 1-11 Longitude and latitude of stars 652 12-24 Naksatra shape, no. of stars, yogatara654 25-40 Other important stars 656 41-56 Saptarsi mandala 659 57-59 Circumpolar stars 666 60-63 Pole stars 667 64-75 Arhsa of kala, mana, ksetra 669 76-79 Sara of naksatra 673 80-87 Conjunction & bheda 674 88-94 Milky way, conclusion 677 13. RISING SETTING OF 682-707 PLANETS, STARS 1-6 Daily and heliacal rising, setting 682 7-11 Drkkarma for heliacal rising 684 12-25 Kalarhsa of moon, stars, planets 687 26-29 Rules for heliacal rising 689 30-33 Days of rising • 691 34-37 Aksa drkkarma of stars * 692 38-44 Udaya dhruva of stars * 697 45-50 Sphuta Kalarhsa 699 x Siddhanta Uarpana 50-58 Sphuta Udaya dhruva 700 59-68 Kalaihsa of planets for Orissa 702 70-82 Udayasta time & end 703 14 - LUNAR HORNS 708-730 TEXT 1-13 Time of rising and setting of moon 708 14-18 Moon at desired time (nata) 713 19-27 Elevation of Horns 714 28-29 Sara valana 715 30-43 Diagram of horns . 719 44-61 Modern method * 722 62-63 Horns of budha, sukra 728 64-69 Reasons of new methods, end 728 15. Mahapata 731-754 1-8 Two mahapata 731 9-15 Calculation 734 16-20 Mean value for current ayanamsa 736 21-33 Pata from sphuta kranti 737 34-42 Sparsa and moksa of pata 742 43-58 Effects, duration 745 59-62 Siddhanta methods 750 63-66 Start of eras 751 67-68 Importance of siddhanta etc. 752 69-73 Dhruva, age of Brahma, end 753 D. GOLADHIKARA !6. QUESTIONS ON METHODS 755-768 TEXT 1-4 Scope 755 5-11 Importance of gola 757 Subject Index 16-20 40-45 46-49 50-56 57-65 66-69 70-78 79-81 17. 18. 1-12 13-32 33-38 39-78 79-93 94-101 102-104 105-111 112-115 116128 129-133 134-142 143-145 147-161 1-2 3-22 23-32 33-38 39-42 43-44 Doubts about earth motion Size of earth, orbit speeds Bhagana revision, guru years True planets, kranti, seasons Eclipse, conjunction Star light, size of brahmanda Meru, mahapata, horns, time units 767 768 769-811 769 XI 759 762 763 764 765 766 Easy methods LOCATION OF EARTH Support of earth Earth as large sphere Bauddha view Motion of earth refuted Planetary motions from fixed earth 791 Sara of planets Vapours of planets Centre of mass Effect of earth rotation Distance & revolution period Manda, sighra kendra Distance, motion of stars Darkness in night God's desire for fixed earth etc. DESCRIPTION OF EARTH Scope 773 777 780 ^794 795 797 799 800 802 803 805 807 812-887 812 Creation as per surya siddhanta 813 Comments on creation 826 Kala of moon, its water 827 Light of stars, sun * 828 Beings on planets 829 847 849 850 xu Siddhanta Darpana 45-47 Composition of earth 829 48-53 Jambu dvipa and meru 830 54-81 Height & vision limit 839 82-87 Sun set at meru 88-91 Location of India 92-98 Variation in oceans 99-108 Meru 851 109-152 Geography from Bhaskara 852 153-167 Dimensions of earth, sphere 876 168-175 Length units, end 885 19. EARTH AND SKY 888-922 TEXT 1-6 Air spheres 888 7-11 Nature of orbits 889 12-21 Orbit lengths 890 22-29 Linear motion of planets 892 30-31 Sara gati 894 32-37 Diagrams for true motion 895 38-42 Less lambana of tara grahas 897 43-52 Larger distance of sun 898 53-58 Revision of orbits and naksatra Kaksa 901 59-69 Measuring distance, diameter 902 70-85 Height of hill, tree, cloud 906 86-102 Vision limit for different heights 913 103-108 Lords of days etc. 918 109-125 Spread of light etc. 919 20. INSTRUMENTS 923-971 TEXT 1-5 Scope 923 6-44 Gola yantra of 2 and 1 axis 929 Subject Index xm 45-75 Multiple axis yantra 937 76-77 Kala yantra 943 78-81 Golardha yantra 944 82-92 Mana yantra 948 93 Cakra, capa, turiya yantra 957 94-97 Time measurements 961 98 Phalaka yantra 963 99-108 Svayamvaha yantra 968 110-113 End 971 21. REMAINING EXPLANATIONS 972-1025 1-6 Scope, day lengths 972 7-13 Day night at poles 973 14-17 Other day nights 974 18-22 Rising time difference in rasis 975 23-26 Ecliptic parts visible at places 976 27-31 Unequal speeds, centre at sun 976 32-41 Calculation from kalpa 977 42-54 Bhagana corrections 980 55-70 Correction after 10,000 years 982 71-78 Guru years 984 79-83 Padaka calculations 985 84-86 Geocentric values 986 87-112 Eccentric of moon 987 113-114 Ucca kaksa 992 115 Correction for moon 993 116-129 Direct and retrograde motion 993 130-142 Sighra and manda gati 997 143-146 Revised methods 1004 147-151 Bhaskara's manda, sighra gati 1004 152-163 Ecliptic, nati, lambana 1005 22. 23. xiv Siddhanta Darpana 164-173 Vitribha „„*, 17/1 ion t^ i 1007 lln'iL A ^ dUrati ° n by dia ^ am 1008 190-192 Ayana valana 1010 lltlll P ° lar riSing ' m °° n horn ' niahapatalOll 199-206 Importance of star circle ion 207-224 Seasons, their zones 1012 225-231 Formation of rain 1014 232-248 Cube and cube root . lm * 249-252 Conclusion ln I, 1024 E. KALADHIKARA SAMVATSARAS ETC 1026-1036 TEXT 1-15 Importance of kala 1026 16-25 Time units from sun and moon 1028 25-36 Nine measures, candra, naksatra 1029 37-46 Solar times 47-60 Sankrantis 1031 61-70 Different years 1033 71-77 Harivamsa units, river tide etc. 1034 PURUSOTTAMA STAVA i 037 (not translated) M- UPASAMHARA 1037 . 1078 TEXT 1-2 ^P 6 1037 3-13 Kautuka panjika 1038 14-29 Corrections to tithi, naksatra, yoga 1042 30 Adhimasa 1047 31-35 Naksatra, rasi of sun 1048 36-38 Sun, moon speeds 1049 Subject Index xv 39-45 Possibility of eclipse 1050 46-57 Tara graha in kautuka panjika 1052 58-60 Pata and mandocca of moon 1058 61-62 Complete revolution of planets 1060 63-65 Soiar dates, use of Kautuka panjika 1061 66-92 Topics in various chapters 1062 93-139 Longitude, latitude of 109 places 1067 140-141 Distance between places . 1073 142-144 Purpose of book 1074 145-149 Author and his family 1074 150-160 Conclusion 1076 24A Appendix (Sanskrta terms) 1079-1094 Name & subject index 1096-1115 INTRODUCTION (1) Arrangement of the book - Scope - Original book was written in 2,500 Sanskrit verses in Oriya script on palm leaves. It was published with introduction in English by Prof. Jogesh Chandra Roy of Ravenshaw College, Cuttack by Calcutta University in 1899. Subsequently same edition was reproduced with approximate Oriya translation by Pandit Vir Hanumana Shastri, by Utkal University, Bhubaneswar (then at Cuttack). This was reprinted by Dharmagrantha Stores, Cuttack. Some parts have not been translated and explained. First volume of this book renders the Sanskrit verses in devanagari script with literal Hindi translation. It also contains the original introduction. . Present second volume translates all the verses in English. Translation is not literal but in mathematical terms, but preserving the technical terms in Sanskrit. Verses in praise of god have been left out, not because of disrespect. With all devotion inspired by Samanta Chandrashekhar, this is not the purpose of the second volume. In addition to translation, each formula has been explained or derived according to modern mathematics and astronomy. The methods have been compared with other Indian astronomers and some times with other countries and with modern astroftomy. This was the method and purpose of Samanta himself. 2 Siddhanta Darpana Technical terms and their calculations cannot be explained in words alone. So a general mathematical and technical introduction is given at beginning of each chapter with bibliography or source reference for further study. In that light only, the methods proved in the chapter can be understood. Where-ever considered useful, methods have also been explained with examples, based on text as well as modern astronomy. In Sanskrit verse, some number or statement has been continued in many verses due to poetic and literal explainations. They have been clubbed together for translation. For brevity and simplicity, many parts have been given in chart form. Chapter 23 contains only verses in praise of god. Most of these verses have two or more meanings. It cannot be expressed in other language, nor it is related to the main topic. It is, therefore, omitted.* 2. Numeration Decimal system of writing numbers originated in India. Arabs called them Hindu numerals. Europeans learnt from Arabs and termed them Arabic numerals. This system uses 10 symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, each increasing by one. For writing greater numbers, successive positions towards left areused, each place having ten times the value of position on its right side. Similarly, fractions are written towards right from the unit place after giving a point, called decimal. Each place has value of l/10th of the value of its predecessor towards left. Modern computers use binary system with two symbols and 1 only, each place value Introduction 3 increasing two times towards left. In angular and time measurements of Indian astronomy, continued till today, multiples or divisions by 60 at each step is used. This was used in Sumerian mathematics for all numbers and is called sexa-gesimal (60) system. Aryabhata, I, has given the following order of place values, each ten times the preceding - Eka (units place), Dasa (ten place), Sata (hundred), Sahasra (thousand), Ayuta (ten thousand) Niyuta (hundred thousand or lakh), Prayuta (ten lakhs or a million), Koti (ten millions or 1 crore) Arbuda (10 crores), Vrnda (100 crores) etc. Sankara Varman in his Sadratnarhala (1,5-6) has given the following sequence in multiples of 10 - Eka (1), Dasa (10), Sata (100), Sahasra (1,000), Ayuta (10,000), Niyuta (or lakh, 10 5 ), Prayuta (10 6 ), Koti (10 7 ), Arbuda (10 8 ), Vrnda (10 9 ), Kharva (10 10 ), Nikharva (10 11 ), Mahapadma (10 12 ), Sanku (10 13 ), Varidhi (10 14 ), Antya (10 16 ) and Parardha (10 17 ) Lalita-vistara, a Buddhist text gives powers of 10 beyond 100 koti (i.e. 10 9 ), each increasing 100 times the previous - Koti (10 7 ), ayuta (10 9 ), niyuta (10 n ), kankara (10 13 ), vivara (10 lS ), aksobhya (10 17 ), vivaha (10 19 ), utsanga (10 21 ), bahula (10 23 ), nagabala (10 25 ), titilambha (10 27 ), vyavasthanaprajnapti (10 29 ), hetuhila (10 31 ), karahu (10 33 ), hetvindriya (10 35 ), samaptalambha (10 37 ), gananagati (10 39 ), niravadya (10 41 ), mudrabala (10 43 ), sarvabala (10 45 ), visajnagati (10 47 ), Sarvasajna (10 49 ), vibhutangama (K? 1 ), taUaksana (10 53 ). 4 Siddhanta Darpana Aryabhata notation - Varga letters (k to m) should be written in varga places (unit place and hundred times at each step) and avarga letters (y to h) in the avarga places. Varga letters take the numerical values (1,2,3 25) from k onwards. Numerical value of the initial avarga letter y is n plus m (i.e. 5+25), next letters are 40 to 90. In nine places of double zeros, nine vowels should be written (one vowel for each pair of varga and avarga letters). Katapayadi notation was before Aryabhata and is believed to have been used in vedas in portions related to astronomy or mathematics. It was very popular in Kerala. Each digit is represented by a consonant letter. Vowels and half letters have no meaning. Digits are written from right to left to form a number. Numbers 1 to nine and the are indicated by letters starting from k, t, p, or y, hence the system is called katapayadi. 1 2 3 4 5 6 7 8 9 k kh g gh n c ch ja jh n t th • d • dh • n • t th d dh n P P h b bh m y r 1 V s s ■ s h 1 Suryasiddhanta and other works including the present book have used words to indicate each digit again written from right to left. These have already been indicated in the 1st volume for purpose of literal hindi translation/ and need not be repeated here. Introduction 3. Transliteration of Sanskrit letters vowels Short 3T ? 3 3s "& a 1 u r • 1 Long 3rfT i 3i ^ atr ^ 4 a 1 6 e o ai au Anusvara • m, m Visarga * h • Consonants ^ n. T \ ^ k kh g •gh n S ^r H ^ c ch * J jh n z 3 ^ • TTf * t ft th • d * dh * n • *t 1 \ H t th d dh n * \ T *t 1 1 p ph b bh m \ ^ ^ *t 1 ■ '* * * y r 1 V s m s • h 1 4. A brief survey of Indian astronomy Astronomy has come from old French word 'astronomie' which in turn was derived from Latin 'astronomia' and Greek 'astronomos' - meaning star law. 'Jyotisa' in Sanskrit means the same - 'Jyoti' means source of light i.e.a star in a sky; study of star groups and motion of planets observed through them is jyotisa. 6 Siddhanta Darpana Greek astronomy had its origin in Nile river and Sumerian civilisation. Western astronomers try to establish that vedic jyotis is originated from Sumer and later Indian astronomy is influenced by Greek. But internal astronomical evidence suggests that text of vedanga jyotisa was written in 2976 B.C. when summer solstice started (verse 6 tells that Magna month began when Sun was in mid Aslesa. Full moon point was 1°13' east of mid magna i.e. 8° east of Regulas at present at 150° 56' - difference of 68° 56'). However, system is much older, and many changes have been made from Taittiriya Samhita. Vedanga jyotisa is found in two texts - Rkveda has 36 verses on the topic and yajurveda has 43 verses. Many are common, but the system is entirely different. Yajur jyotisa was written 624 years after Rk jyotis according to internal evidence. Compiler of Rk jyotis is 'Lagagha.' According to difference in day lengths, mentioned in verse 7 and 22, they refer to a place of 34° 50' North latitude. In northern borders of India, this is near Alma - Ata of Kyrgiz. Since it was first seat of learning, first school is called alma-meter is Greek. So far, authors have assumed that both versions of vedanga jyotisa denote 5 years yuga (or cyclic period). Accordingly text of Rk jyotisa had to be modified and twisted. But now Sri P.V. Holay of Nagpur in his Vedic Astronomy (1988) has proved that the original text of Rk jyotisa indicates a 19 year yuga - after which solar and lunar years start together. There are 7 extra months in a yuga, their adjustment is such that 5 solar years start within 6 days of new moon. Such Introduction / approximately concurrent years are called Samvatsara. Other types of years are Anuvatsara, Parivatsara, Idvatsara and Idavatsara. Thus the statement that a yuga has 5 samvatsaras doesn't mean that 5 years make a yuga as assumed so far. It means only that out of 19 years in a yuga, 5 are samvatsaras. 19 years cycle was later on discovered by a Greek astronomer Metori in 432 B.C. and is called Metonic cycle. However, this cycle was used by Sumerians and Chinese also in their calender much before the Greeks. It is certain that astronomy in the whole world had single system. Irrespective of origins, there was exchange and compilation of ideas, and same standard was adopted as in the modern sciences. Thus houses of zodiac and constellations have the same names in all the languages. There is similar correspon- dance in medical names of Greek origin and their Sanskrit names in yoga or ayurveda. Vedanga Jyotisa was followed by Garga Samhita and Paitamaha Siddhanta and Jain works Surya-pannati and Jyotiskarandaka with minor changes. This period was followed by so called Siddhanta period. According to traditional Indian belief, there were 18 such siddhantas - (1) Surya (2) Paitamaha (3) Vyasa (4) Vasistha (5) Atri (6) Parasara (7) Kasyapa (8) Narada (9) Gargya (10) Marlci (11) Manu (12) Angira (13) Lomasa (or Romaka) (14) Paulisa (15) Cyavana (16) Yavana (17) Bhrgu and (18) Saunaka, Five of these siddhantas - Saura, Paitamaha, Vasistha, Romaka and Paulisa - were codified by Varahamihira in his Pancasiddhantika (184 B.C.) who has^ emphasised that the Saura was most accurate of them. 8 Siddhanta Darpana Saura or Surya Siddhanta has no human authorship. Second verse of the text states that when short time (or 121 years in Katapayadi) was remaining in end of Satyuga, Sun god taught this to Maya asura. Yuga system of this originates from Visnudharmottara purana according to Brahmagup- ta which is modification of old Brahma (or Paitamaha) siddhanta. In Varahamihira's Saura, a period of 180,000 years has been stated which contains 66,389 inter calary months and 10,45,095 ommitted lunar days (tithis). Modern Surya Siddhanta tells about a mahayuga (or yuga) of 43,20,000 years divided into Krta, Treta, Dvapara and Kali ages in ratio of 4:3:2:1 (12,000 divine years) with 1/12* period each in beginning and end as sandhya (twilight period). 360 solar years are called a divine year. Paitamaha siddhanta is crudest and has 5 years yuga like yajus jyotisa. Vasistha has improvement and deals with true motion of 5 planets. Sidereal year has been stated of 365 1/4 days. Paulisa siddhanta is more accurate and gives days counts (ahargana) and sine tables. It gives solar year of 365.2583 days. Al-Barurti has regarded Paulisa as a Greek from Alexandria (Sachau I, p/153). Romaka gives a luni-solar cycle of 2850 years with 1,050 intercalary months and 16,547 omitted lunar days. Length of year is 365 days 5h 55' 12" and synodic month is 29 days 12h 44'2.2". It deals with equations of centre for Sun and mooni Among present compiled texts, Aryabhatiya of Aryabhata I (476 A.D.) of 121 verses is the first. Introduction 9 * It is a brief codification of existing knowledge after observatory (khagol village) near Kusumpur (modern Patna, capital of Bihar) was destroyed in Huna attack. It is more an attempt to preserve the science in verse form, than to write a text book. For brevity, he has devised his own number system, as explained before. Subsequent astronomers made appropriate corrections and devised simpler methods of calculations in their texts. Jyotisa has three parts - (1) Ganita - cor- responding to modern astronomy and mathematical methods (2) Phalita - Astrology (3) Hora or Samhita - auspicious times, natural phenomena, signs in human beings and animals etc. Ganita' is written in three styles - (1) Siddhanta is a text for calculation from beginning of yuga. (2) Tantra starts the calculation from beginning of Kaliyuga (17/18- 2-3102 B.C. Ujjain mid-night) (3) Karana uses short methods for current years ephemeris with reference to a recent base year. Its literal meaning and use is same as that of a handbook or a manual. A brief list of astronomers and their works is indicated below - 5th-6th Century - Aryabhata I (Aryabhatiya and Aryabhatasiddhanta, the later available only in quotations). 6th Century - Prabhakara, pupil of Aryabhata, Varaha-mihira (Pancasiddhantika and Brhatsa- mhita). 6-7th Century - Bhaskara I (Mahabhaskariya, Laghubhaskariya and Aryabhatiya - bhasya); Brah- magupta (Brahma-sphuta-siddhanta and Khanda- 10 Siddhanta Darpana khadyaka), Haridatta (Grahcara-nibandhana) Devacarya (Karana-ratna). t 8~9th Century - Lalla (Sisya-dhi-vrddhida- Tantra) Govinda - Svamin (Mahabhaskariya-bhasya) Sankaranarayana (Laghu-bhaskariya vivarana) Prthiidaka svamin (Brahma-siddhanta vasana bhasya) and Khanda-Khadyaka vivarana. 10th Century - Vatesvara (Vatesvara- siddhanta) Munjala (Laghumanasa), Sripati (Siddhanta-sekhara) Aryabhata II (Mahasiddhanta), Bhattotpala (Khanda-Khadyaka vyakhya and Vrhatsamhita - vyakhya) Vijayanandin (Karana Tilaka). 11th Century - Somesvara (Aryabhatiya Vyakhya) Satananda (Bhasvati) 12th Century - Bhaskara II (Siddhanta Siromani with Vasana bhasya, Karana Kutiihala), Mallikarjuna Suri (Surya siddhanta Vyakhya) Suryadevayajvan (Aryabhatiya Prakasika and Laghumanasa Vyakhya) Candesvara (Surya- Siddhanta Bhasya). 13th Century - Amraja (Khanda-Khadyaka- Vasana bhasya) 14th Century - Makkibhatta (Ganita Bhusana), Madhava of Sangamagrama (Sphuta candrapti, Aganitagrahacara, Venvaroha), Madanapala (Vasanarnava on the Surya siddhanta), Viddana (Varsika Tantra), 15th Century - Paramesvara (Drgganita, Goladipika) Grahanamandana, Grahana-nyaya- dipika, Aryabhatiya vyakhya, Bhatdipika, Mahabhaskariya vyakhya, Laghubhaskarlya vyakhya, Surya siddhanta-vyakhya and Introduction ** Mahabhaskariya bhasya vyakhya) Yallaya (Aryabhatlya vyakhya, Jyotisa darpana, Laghumanasa-kalpataru arid KalpavalK on the Surya-siddhanta), Rama Krsna Aradhya (Surya siddhanta Subodhini) Cakradhara (Yantra Cintamani) Nilakartha Somayaji (Jyotirmlmansa, Golasara, Candracchayaganita, Siddhanta Darpana, Tantra Samgraha and Aryabhatlya bhasya). 16th Century - Jyesthadeva (Yuktibhasya, Drkkarana) Sankara Variyar (Karana sara, Tantra- sangraha, Yukti dipika), Bhudhara (Surya siddhanta vivarana) Tamma yajvan (Grahanadhikara, Surya siddhanta - Kamadogdhri), Ganesa Daivajna (Grahalaghava, Tithi-Cintamani, Pratodayantra and Siddhanta Siromani-Vyakhya), Acyuta Pisarati (Karanottama, Sphutanirnaya with vivarana, Uparagakriya - Krama, Rasigola sphutaniti); Rama (Rama vinoda) 17th Century - Visvanatha (Grahanartha Prakasika, Grahalaghavatika, Karanakutuhala Udaharana) Candidasa (Karana Kutuhala TTka), Putumana Somayaji (Karana paddhati, Pancabodha, Nyayaratna) Nityananda (Siddhantaraja and Siddhanta sindhu) 18th Century - Maharaja Sawai Jayasimha (Yantraraja racana, Jayavinodasarani), Jagannatha Samrata (Samrata Siddhanta) 19th Century - Sankaravarmana (Sadratnamala) For easy calculation of pancanga, many astronomical tables have been prepared. These are called Kosthaka or Sarani. Early examples are Grahajnana by Asadhara^ (epocb-20-3-1132), Laghukhecara siddhi by Sridhara (20-3-1316), 22 Siddhanta Darpana Makaranda by Makaranda (epoch 27^3-1478) Kheta- muktavali by Nrsimha (31-3-1566) (5) Astronomers of Orissa Orissa was part of the Indian tradition of Jyotisa from vaidic and siddhanta period. Astronomy and mathematics were related to Yajnas whose time was found with astronomy and construction was as per geometric diagrams. In Orissa, Brahmanic titles related to yajnas still exist like - Hota; Udgata, Brahma, Pathi, Pari, Vagmi etc. It is quite probable that Taittiriya Samhita and Aranyaka, Aitareya and Gopatha Brahmana etc. - the aranyaka granthas forming origin of astronomy flourished in places like western Orissa which were famous as aranya or mahakantara. Another indica- tion of rise of astronomy is the sea trade from Orissa coast to East Asia and upto Roman Empire. Due to popularity of Bali yatra, it is thought that sea trade of Orissa was only with Bali - a small island in Indonesia. However, the relations must have developed with other areas of South East Asia and Chinese coast and intermediate islands of Andamana group must have formed base for supply of food etc. Late Dr. N.K. Sahu in his history of Orissa states that silk of Sambalpur was known in Roman empire also. This confirms that ships from Orissa and other parts of India were going to different parts of the world. Technically, visit to America was also possible and traditional jyotisa texts mention a town 90° east of Ujjain (yamakoti) which should be in New Zealand (southern hemisphere). Hence yama is lord of south direction. 180° east of Ujjain in Siddhapura in North hemisphere. At this longitude there is a Introduction x ^ town near Mexico where greatest Pyramid was built - Valmiki Ramayana calls it a gate built by Brahma to indicate end of east direction i.e. 180° East of Ujjain at prime meridian in Indian Astronomy (Kishkindha kanda). To a layman this discussion appears irrelevant to astronomy. However, sea journey (and plane journey in modern times) is not possible without knowledge of astronomy. There are no landmarks in sea or sky for finding the way. Hence navigation requires accurate deter- mination of longitude, latitude and direction. These three are discussed in an important chapter Triprasnadhikara' of Surya siddhanta. It is note- worthy that Columbus could undertake his journey in open sea only because method of finding longitude was discovered in western astronomy ten years before that. That was from Turkish ships who had learnt astronomy from India. Vice versa, longitude determination in remote past indicates that India was well versed in navigation round the globe. Transport of rice from Orissa was marked by Salivahana Saka in 78 A.D. - it means transport of rice (Sali = rice, Vahana - carriage). As a product of Audra (Orissa), rice was called Audriya i.e. Oryza in greak. This has become rice in English (omitting '0') and Orissa as name of the state. Navigation history indicates traditional study of astronomy in Orissa. Surya Siddhanta has been given by sun god, whose worship is most common in coastal areas and river ports in India and elsewhere (Japan, Egypt, Mexico, Peru etc.). Jyotisa study might have suffered during Buddhist era in Orfesa. It again 14 Siddhanta Darpana picked up after Varahamihira in orissa like other parts of India. Gariga period (650 to 900 A.D.) records of Orissa indicate that Brahmans were well versed in Vedanga of which jyotiSa is a part. One person has specifically been mentioned as siddhanti. Satananda was most famous of old astronomers of Orissa. Satananda - He was son-of Sankara and Sarasvati of Purushottampura (Puri) who completed his famous work Bhasvati"' in 4200 yugabda (1099 A.D.). He has made calculations with reference to Purl. Full name of Bhasvati was Panca siddhanta sara or Pancasiddhanta - Bhasvati on pattern of Pancasiddhantika of Varahamihira. However, he has followed Surya siddhanta only which is considered most accurate. It is a Karana grantha following solar year starting from Sayana mesa samkranti. It was popular for its accurate calculation of eclipse - UlpJt*ii*qal«RJT Commentaries on Bhasvati- (1) Sansaraprakasika by Kasisekhara (2) Balabodhini TIka of Bhasvati in 1543 A.D. by Balabhadra son of Vasanta of Kausika gotra in Uma town of Jumila state. (3) Oriya translation by Trilocana Mohanti in Yugabda 4747. Other books of 'Satananda are - (1) Satananda Ratnamala - a samhita book like Ratnamala of "Sripati, his elder contemporary. (Palm leaf manuscript No 268, Orissa Museum). (2) Satananda Samgraha - work on smrti. No manuscript is available. Only Bhasvati is available with Hindi com- mentary by Matr Prasada Pandey by Chaukhambha, Varanasl. Introduction 1$ Other astronomers of Orissa - (1) Jayadhara Sarma of Kotarahanga near Sakhigopala (Purl) received grants from Bhanja kings in 1231-12;* 5 A.D. for his mastery on Jyotisa. Though he was famous, no book by him or his forefathers is available. (2) Gajapati Kapilesvara Deva (1435-1466 A.D.) of Cuttack who started Kapila era got another book written after his name called Kapila Bhasvati. But no manuscript is available. (3) Govinda Dasa of Nagesa gotra son of Hira Devi was a great astrologer. He constructed a dola-mardapa in sacred town of "Sri Kurma". No work by him is available. (4) Trilocana Mahanti - He translated Bhasvati in Oriya verses in 4747 yugabda (1646 A D). (5) Gajapati Narayana Deva of Parla Khemun- di wrote Ayurdaya Kaumudi in 26 chapters around 1650 A.D. (6) Vipra Namadeva - He wrote a samskrta commentary Sarvabodhini on Suryasiddhanta in 1721 AD. (7) Dhananjaya Acarya - wrote a Palaka Panjika for 1665 Saka (1733 A.D.). 18 chapters of his Jyotisa candrodaya are available in Orissa museum. He wrote another work Jataka Candrodaya. (8) MagunI Pathi, son of Markandeya Pathi wrote an Oriya commentary Mandartha bodhini on Siddhanta Siromani in 1741 A.D. In 1744 A.D. he wrote another commentary in Oriya on Grahacakra of Kocanacarya. m 16 Siddhanta Darpana There is an incomplete work Jyotis Sastra by Markanda who may be his father. (9) Mahamahopadhyaya Dayanidhi Nanda wrote Sisubodhini in 1707. (10) Mahamahopadhyaya Chapadi Nanda wrote Balabodharatna Kaumudi in 1763. (11) Son of Srinivasa Misra wrote Jyotis tattva Kaumudi in 18th century. First 12 chapters are available. (12) Gadadhara Pattanaik S/o Padmanabha in 18th century wrote Ravindu grahanam on basis of Kocana-carya in 18th century. (13) Gopinatha Dasa (Patnaik) wrote Ayurdaya Siromani and Suddhahnika Paddhati. (14) Caitanya Raja Guru - wrote Laghusiddhanta on pattern of Surya Siddhanta and wrote one Oriya commentary on it. (15) Yajna Mishra S/o Visyambhara wrote Jyotisa Cintamani or Ratnapancaka whose incom- plete manuscripts are available. (16) Mahidhara Mishra wrote Mahldhara Samhita in 18th century and a commentary on Amarakosa. (17) Prajapati Dasa - (Unknown time) - Grantha Samgraha, pancasvara and Saptanga. (18) Bhanusekhara Dasa (18th century) Tarani Prakasika , a commentary on Jataka Ratnakara. (19) Dasarathi Mishra (18th century) - Jyotisa Samgraha. (20) Krsna Misra (18th century) - Naksatra Cudamani, Kala Sarvasva. Introduction 17 (21) Tripurari Dasa - Oriya poet of 17th century - He wrote the following books on Kerala astronomy - Kerala Sutra, Kerallya dasa and Prakrta Kerala. (22) Nilakantha Praharaja and his son Yogi Praharaj - Their books Smrti Darpana and Vaidyahrdayananda have been published by Madras Govt. (6) Samanta Candrasekhara and his role Brief Biodata - He was born on 11.1.1936 (Tuesday) i.e. Pausa Krsna 7/8th 1892 Vikramabda (1757 saka) For an astronomer it is proper to give his birth time by planetary positions which is free of a calendar system. Birth time - 09-04 1ST based on Kumbha lagna and dasa calculation Birth place - Khandapara (Purl ) Latitude 20° 15' North, longitude 85°6'East Lagna 310° 40' (Pranapada in 5th house, Navamsa is Makara. Ayanamsa 21°34' Sun 268° 28' Moon 172 Q 35' Mars 263° 26' Mercury 271° 45' Jupiter 77 '40' Venus 292° 25' Saturn 192° 46' Rahu 34° 56' Uranus 306° 59' Neptune 281° 31' Balance of Moon dasa of birth - 6 months 23 days. Important events of his life - He did not have formal university education. Even though he was born in a royal family, he sufferred poverty ^ Siddhanta Darpana and unhappy family throughout his life. He sufferred from chromic dyspepsia and stomach inflammations frequently. At the age of 22, he married princes Sita Devi of Angula Raj family. Due to his ugly looks his father-in-law showed reluctance to give his daughter in marriage in lagna mandapa. When he showed his deep knowledge of Sastras and mastery over Samskrta verses, his marriage was solemnised (possibly on 28-2-1858). He had 5 sons and 6 daughters, out of which two sons had expired. He was banished from Khandapada by his ruler, being his own cousin. But due to his knowledge, he gained wide fame and his rights were restored by the then commis- sioner of Cuttack. He was also given a 'Sanad' in honour of his achievements. His work "Siddhanta Darpana" cannot be fully understood by a person unless he is well versed in Indian astronomy as well as modern mathematics. Whatever is known to common public about the book or its author is based on the English introduction by Prof. Jogesha Chandra Roy. This is based on personal interviews and not on a stfudy of the book. So, many vital points have been left out. Samanta expired on 11.6.1904. On the basis of -his horoscope he had foreseen his death; which is expressed by his son Gad&dhara in an Oriya verse-meaning - "father called me near and told that moon had entered his maraka naksatra, and there was no escape from death". In last but one verse of Siddhanta Darpana, he has expressed desire that his body should fall at the feet of Lord JagannStha. On his last day, he went for darshana of lord. At the time of bowing before Jagannatha, he expired. At every place in Introduction ^ the book, he has shown his deep faith in lord and the scriptures. He has accepted his experimental observations only when they found support in some scripture. His works - Siddhanta Darpana is work of his whole life. At the end of every chapter two fold purpose of this book is explained - (1) Balabodha - i.e. a text book and (2) Ganita-Aksi Siddhi - i.e. tally of calculation and observation. For text book purpose, this is a treatise on Indian astronomy containing relevant positions of all text books from Sakalya Samhita to siddhanta books starting from Aryabhata. Quotation from Atharvaveda is unique in Indian astronomy; as it is only correct figure for sun's diamater,in Indian astronomy or in western astronomy before advent of telescope about 300 years ago. It is most voluminous book on astronomy with 2500 verses. Next largest are Vateswara Siddhanta with 1100 verses or Siddhanta Siromani with 900 verses. Ganita-Ak§i Siddhi has three fold significance. As every other science, purpose of astronomy is to tally mathematical calculations with observations. Books starting with Aryabhata have only formulated or coded the existing knowledge, they have not indicated source of such figures. Methods are often in-completely explained and only refer to Vedic origin which is not clear. Thus purpose of math is only to find calculation methods for finding the observed position. Siddhanta books are not concerned about mathematical models, theories of gravitation or theories of motion. We are satisfied 20 Siddhanta Darpana when calculations give correct result and not bothered whether Sun or earth is centre of motion. Third aspect is that there is slight change in planetary motion over long periods of time as stated in Surya siddhanta. This happens due to tidal friction. But siddhanta texts after Aryabhata have assumed constant motion throughout yuga or a kalpa of 1000 yugas. Due to approximation of constants or errors in calculation methods there is some deviation in observed results. In every period astronomers have corrected the constants given in Surya Siddhanta. according to need. These are called Bija corrections. Researches of Candrashekhara : (1) Moon's Motion (a) Traditionally moon's equation was of the form - 300'49.5," Sin (nt-a) + 2'23.25" Sin $. (nt-a) 2nd term is equation of apsis introduced by Brahmagupta. This form is correct, but constant is slightly wrong. (b) Sripati had found effect of Sun's attraction on moon motion (called evection). This has been introduced as Tungantara correction by Candrasekhara given as -160' cos(0 -a) Sin (D - 6) X Moon's apparent daily motion daily mean motion Error is about 4' only. (c) Bhaskara II had observed a fortnightly variation in moon's motion giving an error of maximum of 6 dandas in middle of pak§a. Introduction 21 Comparing with his own observation, Samanta gave the Paksika equation as 38'12" Sin 2 (D-d) where D' is moon corrected by 1st and 2nd equation (d) Due to Sun's annual motion, a digamsa correction also was introduced. ±11'27.6" Sin (Sun's distance from apogee) These equations almost give the modern value and are to be further checked after 1000 years. (2) Ayanamsa - According to modern theory, earth's axis is rotating in a conical motion complet- ing almost uniform circular motion in 25,726 years. Samanta has assumed libration theory that the motion is not circular with 360" rotation but pendulum like oscillations within values of 27 ° , but with uniform motion. Present value of Ayanamsa tallies with both theories. Only after 300 years or so error may be noticed. He has corrected the value of Surya siddhanta slightly (6,40, 170 revolutions in a kalpa, instead of 6 lakh revolutions according to Surya siddhanta). Liberation theory is not supported by modern astronomy but it may be correct according to methods of projective geometry used in Jain Astronomy (Thesis by Sri S.S. Lishk). (3) Mandocca gati - According to classical mechanics, planets move in elliptical orbit whose major axis is fixed in space. Partly due to acti of other planets (mainly jupiter) and partly due to general theory of Relativity (1917 - Einstein), force of attraction reaches at speed of fight, not 22 Siddhanta Darpana instantaneously - mandocca (apogee) is moving slowly. For mercury, it was calculated to be I* in 11,000 years which was tested in 1919. In 300 years since Tycho Brahe, it is only 1/36° of an angle. For other planets it is so slow that it cannot be measured even by modern instruments. Indian astronomy gives V in 12,000 years for mercury and 39 revolutions of Saturn in 1 Kalpa or 1° in 3 lakh years. 'Cosmology' by Narlikara gives its rate of movement as 6 Jt G M L Tc 2 where M = mass of Sun, T = period of planet Samanta has introduced a new concept of Parocca for Mars and Saturn which moves with constant circular motion around which mandocca oscillates. This is not supported by relativistic equation. But it may be probable due to effect of Jupiter between Mars and Saturn, which can be tested only by a computer calculation. Another doubt is that such a motion cannot be observed in one life time. Even Moon's equation of motion is based on 1000 years of observation and needs same time more to test it. Samanta has not mentioned the basis of his correction. (4) Discussion of other theories - Prof. J.C. Ray had not read Siddhanta Darpana and wrote introduction on basis of personal discussions. But Samanta has treated him as student and has criticised his opinious about modern physics in his chapters on discussions (Vasana-rahasya). 23 Introduction (a) Jain theories have been criticised because they were based on projective geometry and become absurd according to spherical geometry. As a single* sphere of earth is drawn as two circular maps in projective geometry, two Suns and two Moons were assumed in Jain theories. But dimen- sions of imaginary mountain 'Meru' have been quoted on the basis of Jain theories only. (b) In Indian astronomy, for calculation pur- pose, it is immaterial whether earth in fixed or it moves. In both views, relative motion will be same riving the same result. Samanta has used modern physics to refute the theory that Sun is not the centre of motion. It is mass centre of solar system, which is away from Sun's surface at a distance of l-l/2times its radius in direction of Jupiter (effect of other planets can be neglected). It appears that, Samanta was too skeptic of European theories whom he has called 'golden theory' as they were supported with hope of getting gold medal (17-160) (c) His other objection was that if earth moves on its axis; why Jupiter moves faster being the heaviest. This has been explained later on by presuming that Jupiter and Sun were twin stars. Due to loss of matter, Jupiter gained m angular speed, to preserve the momentum. The other objection as to why we observe the same side of moon - has not been explained so far. (d) If stars are all like Sun and are equally spaced in all directions, there should be no day and night - every time equal light should come from all directions. This was called Olber's paradox in modern astronomy and was explained only m 1930 when expansion of universe was observed (It is mentioned in Indian scriptures also) Due to expansion, the farther stars have lesser effect and 24 Siddhanta Darpana only the Sun causes day and night. Samanta has correctly refuted the argument of absorption of light of stars by gases etc. (5) Diameter of Sun : Diameter of Sun had been heavily under estimated to be about 10 to 14 times the diameter of moon by all astronomers in India and outside. After talescope it was known to be 400 times. Distance of Sun being 400 times that of Moon, it will cause much greater difference in amount of solar eclipse at two places. This might have prompted Samanta to correct it. But he has referred to Brahmavidya upanisad and Atharva veda (describing expansion of 3*) to get the value of 72,000 yojans (19-40,50) and (8-12). This become 162 times diameter of Moon. Siddhanta Darpana has taken value of 1 yojans as 4.9 miles. Had. he taken it to be 11 miles (Aryabhata 7.5 miles, Jain theories 9.2 miles) as it was in vedic times, he would have got correct value of Sun's diameter. This shows the absolute faith of Candrasekhara in ancient scriptures without which he never confirmed any result. To some extent it was justified, as seen from correct assessment of Sun's size in vedas compared to all ancient measures. Some of his observations may appear biased or excessive, but they show a marvelous grasp of modern physics. Some of the points were not properly understood by top astronomers of his time REFERENCES 1. Bharatiya Jyotisa Sastra - by Sankara Balakrsna DIksita - Original written in 1896 in Marathi English translation published in 1968 in two Introduction * 25 volumes by Publications division, Govt, of India. Translation in English by Sri R.V. Vaidya. Hindi translation by Sri Visvanatha Jharakhandi published by govt, of U.P. from Lucknow. 2. History of Astronomy in India - Editiors S.N. Sen and K.S. Shukla. Published by Indian National Science Academy, New Delhi-2 in 1985. 3. Indian Astronomy - A source book compiled by B.V. Subbarayappa and K.V. Sarma. Published by Nehru Centre, Bombay - 18 in 1985. 4. History of the calender by M.N Sana and N.C. Lahiri Published by Council of Scientific & Industrial Research, Rafi Marg, New Delbi-1 in 1992. 5. Vedic Astronomy by Sri P.V. Holay, Nagpur - 12 in 1989. 6. Commemoration volume, 1990 - Directorate of Culture, Govt of XDrissa. * 7. Introduction to cosmology - Cambridge, University, The Structure of the Universe - Oxford University Both by Sri Jayanta Visnu Narlikara. 14.4.1997 Arun Kumar Upadhyay Ramanavami Cuttack MADHYAMADHIKARA Madhyama = mean. This portion deals with average or mean motion of planets. Calculation of mean position is done from beginning of Kalpa or yuga for siddhanta, from Kali beginning in a tantra and from epoch of this book (12-4-1869 Monday - 1st saura Caitra 1791 Saka) as a Karana book. Siddhanta Darpana explains all the three methods and in addition last chapter gives easy method for (Calculations of pancanga. There are 33 mathematical tables in the end for ease in calculations.. The book is in two halves. First half deals with the (ganita) methods of astronomy, 2nd half deals with explanations and discussions and special topics (Gola) First half of the book contains three parts called Adhikaras. First part is madhyamadhikara, with 4 chapters (called Prakasa), Part 2 is Spastadhikara with 2 chapters. Part 3 is Triprasnadhikara with 9 chapters. Second half is called gola and has two parts. Part 4 is Goladhikara containing 6 chapters. Part 5 is Kaladhikara containing 3 chapters. First chapter of part 1 is called kala varnana explained below. Chapter - 1 MEASUREMENT OF TIME 1. An Introduction to the Units of measurements Any natural science involves theories and experiments which verify each other. We test the theory by measuring certain quantities and see whether they are according to the theory. The deviations or errors cause refinement in the theory. As in physical sciences (particularly mechanics), the units of measurement in astronomy are of length, degree and time. Basic units in physics are of length, time and mass. Degree is a dimensionless quantity because it is ratio of length of arc to length of radius. Practical units of quantities are based on human experience. Length is similar to hand or feet length, mass is mass of rice measured by spread of palms, time units is based on breathing time of human beings. , . However, standardisation of length units is based on dimension of earth or comparison of some light wavelength. Similarly time units are fixed according to rotation periods of sun and moon or more accurately time taken by light to travel a particular distance. We can see that in modern physics as well as in ancient India *standardsation method was exactly the same. Units of angle also 25 Siddhanta Darpana are based on the number of days (about 360) in a year, and hence sexagesimal (divisions by 60) system was more convenient. 2. Units of length In British system of units, foot was the basic unit equal to average length of human feet. In old Greece and Rome; cubit (18" = one hand) and stadia were also based on human measurements. For smaller units, angula (finger width) was the basic unit in India (0.75" or 1.88 cm). In Tiloya Pannati (Jain Text), 1 angula = 8 9 Trasarenu In Anuyogadvara Sutra ("), 1 angula = glO // In Siddhanta Jyotisa (Sripati), 1 angula = 8 6 trasarenu Successively smaller units of Siddhanta are Angula - yava - yuka - liksha - Balagra - Renu - Trasarenu. Balagra (hair end) is Angul -s- 8 4 = 1/4 x 10 4 crri (micron) Thus the dimensions are really correct has hair is 3-4 micron wide. According to Tiloyapannati lowest division is 1 paramanu = 1 angula (1.88 cm) x 8* 13 cm. = 3.5 x» 10" 12 cms. This is of the order of nuclear diameter. In Lalita vistara (Buddhist text), units are divided by 7 at each stage. According to it, 1 paramanu = 1 angula (1.9 cm) x 7 10 cm = 0.66 x 10 8 cm. This is equal to the Bohr rodius of Hydrogen atom. Measurement of Time 29 Larger units are multiples of angula or a 'purusa' or person (about 6 ft height). It is same as 'fathom' used to measure depth of sea or river. Bigger units in Tiloyapannati are - 6 Angula = 1 pada (foot) 2 pada = 1 vitasti (span) 2 vitasti = 1 hasta (forearm or cubit) 2 hasta = 1 rikku or kisku 2 Kisku = 1 danda (staff) or dhanusa 2,000 danda= 1 Krosa 4 Krosa = 1 yojana Same units have been used by Paulish Siddhanta, Srlpati and subsequent siddhanta texts. Lalita vistara, however makes 1 kosa = 1000 dhanusa only equal to 1/2 Jain or Siddhanta yojana. In the time of Napolean, attempt was made to link length unit 'metre' with dimensions of earth. So 1 metre was proposed to be 10" 7 of distance between equator and north pole. Subsequently, it was learnt that it was 1 crore and 486 parts of this distance. Still, the standard length of platinum bar kept at Paris is used as metre. Nautical mile is also based on earth's dimension but it is not a decimal fraction. It is length of 1 minute of arc at equator (about 6080 ft. or 2 kms) In same way yojana has been defined to be an exact fraction of earth's diameter or circum- ference in polar circle. Varahamihira - Circumference 3200 yojana Aryabhata - Diameter = 1050 yojana «■■ Surya Siddhanta - Diameter = 1600 yojana 30 Siddhanta Darpana Siddhanta Siromani - Circumference = 4,800 yojana (This is followed by Siddhanta Darpana also) Thus, yojana is 5 miles according to Siddhanta Shiromani and 7.52 miles according to Aryabhata. Anuyogadvara Sutra (Jain) gives 1 Atma yojana = 7,68,000 angula = 9-1/11 miles estimated according to current measurements of earth. Dr. L.C. Jain opines that 1 Pramana yojana is 500 Atma yojana = 4,5 45.45 miles. M.B. Panta opines that 5 yojana (40 or 45.5 miles) was called Mahayojana used for measuring distances of stars. For example, 'Trisanku' star is named on basis of its distance from earth. Tri-Sanku = 3 x 10 13 Mahayojana = 207 light years. This is actually the distance of that star now known as Beta-crucis in Southern cross constella- tion. Similarly, it is said that Agastya had crossed Varidhi (10 14 ) or drunk ocean and had gone south. It is now known as 'Argo-Navis' star at 80° 5' south latitude, indicating naval journey. This star is 652 light years away ; 10 14 mahayojan is about 690 light years. Prof. S.S. Dey of Calcutta has observed that Egyptian names of planets mercury, venus, mars, jupiter and saturn give their distances from Sun in yojana if names are interprated in Katapayadi system. At present metre is defined as 16,50,763.73 times the wave length of radiation of Krypton-86 isotope for transfer of electron between 2p and 4 d states. With accurate measurement of velocity Measurement of Time 31 of light, it is proposed to link time and length units. In fact truti (a unit of time) was also defined as time taken by light to travel 1 ybjana. 3. Measurement of Time Principle of time measurement is to choose a unit which is equal to the time of a periodic event (which repeats itself after fixed intervals). Examples of such events are - vibration of quartz crystal or metal spring, pendulum (all used in clocks), rotation of earth (l^day = 24 hours), synodic rotation of Moon (1 month) or apparent rotation of Sun around earth (1 year). Basic unit of time in jyotisa is 'asu' (meaning mouth) or 'prana' (breathing) as it is approximately time (4 seconds) taken by a man in breathing in and out. Since our mental feeling of time is based on breathing only, units bigger than asu can be felt and are called 'Murtta' (tangible). Smaller units are called Amurtta (imaginary). Astronomically, it is time taken by earth in its daily motion (360° in 24 hours) to move by 1' (=1760). 'Amurtta' or small units - In ganita-Sara- Samgraha (Jaina) 1 prana has been divided into 44466-2458/3773 Avalikas. Possible reason for such peculiar ratio is that a muhurtta (1/30 of a day of 24 hours) was equal to 3773 pranas in one system and 1,67,77,216 avalikas in another system. A solar day is divided into 60 danda or ghatika (like 24 hours). Each ghati is divided into 60 pala (each 24 seconds) which is again divided into 60 vipalas. Thus 1 asu or prana is equal to 1/6 pala or 10 vipalas 32 Siddhanta Darpana 1 asu (respiration) = 5/2 kastha 1 kastha = 4 long syllable (gurvaksara) 1 gurvaksara or vipala = 9/2 nimesa (twinkling of eye) 1 nimesa = 100 lava 1 lava = 100 truti (1 Truti is time taken by a sharp needle to pierce a soft lotus petal) 1 Truti = 3 Trasarenu 1 Trasarenu = 3 ami 1 Ami = 2 paramanu TU - T ^ 1 asu (4 second) Thus 1 Truti = — — * — — - — 10 x 9 x 50 x 100 In this time, hight will travel about 2.68 Kms (1/3 or 1/4 yojans or 1 krosa approximately). 1 paramaru Kala = 1/8 Truti = 5 x 10" 7 seconds approx. Larger Units - 10 gurvaksara or Vipala = 1 prana 6 prana = 1 pala or vighati 60 vighati = 1 ghatika 60 ghatika or danda = 1 day (24 hours) 30 days = 1 month (approximate time from one full moon to the next) 12 months = 1 year (approximate time of apparent rotation of Sun) 360 years = 1 divya varsa (divine year) 43,20,000 years = 1 yuga 72 yugas = 1 manu 14 Manu = 1 Kalpa (day of Brahma) Aryabhata Measurement of Time 33 Thus in this system 1008 yugas make a kalpa. Suryasiddhanta gives 1000 yugas in a kalpa with 14 manus of 71 yugas each with 15 sandhis of 1 satyuga (4/10 yuga) each. 2 kalpa = 1 ahoratra (day-night) of Brahma 30 days of Brahma = 1 month of Brahma 12 months of Brahma = 1 year of Brahma = 7,25,760 yugas (Aryabhata) or 7,20,000 yugas (Surya Siddhanta) 100 years of Brahma = Life of Brahma (Mahakalpa or Para) 50 years is called Pararddha = 1.5 x 10 17 years In one mahayuga there are 0.4xl0 17 asus Hence 10 17 is called para or parardha. Concept of yuga - 'Yuga' of Rkveda was of 19 years after when mutual motion of moon and sun repeats itself. Later on, this period was called metonic cycle in Greece. Yajur jyotisa gave a yuga of 5 years which is a simpler system of tallying lunar and solar years. In vedanga jyotisa 19x8+8 = 160 years was next bigger yuga after which lunisolar calender tallies more accurately. Visvamitra had smaller yuga of 3339 tithis = 111 synodic months + 9 tithis. This was half of Saros cycle of Chaldea (223 synodic months or 18 tropical years and 10.5 days) after which ellipses are repeated. His greater yuga was of 3339 synodic years or 3240 sidereal years. One third of this period 108(0 sidereal years was used in determining Indian Eras. This gave rise to small chaturyuga of 4x1080 = 432p years. One Mahayuga is JL0O0 times this unit and 1 Kalpa is 1000 mahayuga. This is 54 Siddhdnta Darpana based on astronomical hymn of Visvamitra (RV III 9-9) - 3339 dyus (days/ tithis/parts of sky) wor- shipped Agni (Sun) by revolutions in the sky. This concept has been used for divisions of constellation in vedanga jyotisa. Siddhanta texts have formed a Mahayuga in which all the seven planets Sun and Moon and 5 faint (Tara) planets make complete revolutions. After a yuga they come to the same position. Thus a position of these planets will occur only once in a yuga and is most accurate method of indicating a time in a yuga. This is one of the purposes of preparing a horoscope. Rotation of mandocca (apogee) of planets is still slower and their full rotations are completed only after 1000 yugas or a Kalpa. Slowest is sani whose mandocca makes only 39 rotations in a kalpa. It may be mentioned that a period of kalpa of 4 bilion years is approximately same as life of earth or the solar system. 2 kalpa or 1 day/night of Brahma is considered to be the time from when universe is expanding and will contract again. Life of Brahma 3xl0 17 years is approximately half life period of proton decay^after which basic elements of the universe will dissolve themselves. 4. Other examples of accurate measurements - Verses of Veda composed by known astronomers Visvamitra, Atri, Sunahsepa, Hiranyastupa, Kutsa, Utathya, his son Dirghatamas, his son Kaksivat and daughter Ghosa - should be read according to Katapayadi system for their mathematical meaning. Measurement of Time 35 Nasadlya and other verses of these sages indicate theories of creation of universe which are similar to modern cosmology. Rkveda (1-164-2) tells that the seven join the body in constant circular motion of earth (ratham). Orbit round Sun is elliptical (called Trinabhicakram) because elipse has 3 nabhis (1 centre and 2 focus) Cakram = 2x^= 6.283, ratha = 72 Hence in Krosa units (= 2.5 miles) 7 x Cakra x ratha = 6.283x7x72x2.5 = 7915 miles which is diameter of earth. Second line indicates Sapta (7) nama (50) vahati (moves in orbit). If movement is taken in 1 lava muhurta = — = 48 seconds. 60 then drbital velocity of earth is 7 x 50 Krosa 7 x 50 2.5 miles 1 lava 48 seconds miles/sec. Tri (3) nabhi (40) Cakra (2rr = 6.283) gives acceleration due to gravity if length unit is taken as hasta 19.8" and time as lava. For small units both are divided by 60. Vilava = 4/5 sec, 1/60 hasta = 0.33/12 ft. 0.33 g = 3 x 40 6.283 x —— x 4 2 = 32 ft/sec 2 Fourth line gives arc of an imaginary sphere on which moon moves. 36 Siddhdnta Darpana Yatra (21), visva (44), Bhuvana (44) gives 21x44x44 = 40, 656 when unit is moon's distance -s- radius of earth. DIrghatamas gives a theory of star formation in RV (1-164-8) - Mata (steller cloud formed by hydrogen atoms) absorbs light (garbharasa) and is further excited by gravilational contraction (pitaram). Dhiti (69) manasa (708) gives diameter of hydrogen atom if we take unit of length 60x60 times smaller than 1/60 hasta (0.33 inches) Dhiti x manasa 0.33 1 X rrr X 2.54 cm. 60 X 60 60 x 708 = 0.478 x 10" 9 cm = radius of hydrogen atom when it is divided by mata pitaram (65x261), this gives 2.8xl0* 13 cm, the distance at which nuclear interaction works. Atomic radius divided by Sa (7) garbharasa (7243) gives 10" 13 cm which is diameter of proton or electron. Velocity of light - Smrtisastra tells - we salute with our respect to sun who traverses 2202 yojans in 1/2 nirhisa. In purana - 1 nimisa = 16/75 seconds In Luavatl, 1 yojana = 4x8000 cubits = 9.09 miles Hence velocity of light is 9.09 x 2202 „ ^ . ._ , = 1.86 x 10 5 miles/sec. 8/75 Bhaskara nimisa is 8/90 seconds, Manu's yojana is 4 Krosa of 4000 cubits each. Then velocity is 3xl0 5 km/sec. 5. Measurement of angles - Since apparent revolution of Sun around earth in a year is in about 360 days, a circle is divided into 360 ° degrees Measurement of Time 37 (amsa) so that motion in one day is about 1°. Its average motion in 1 month (lunar node to node) is about 30° hence 30° is 1 rasi. Further sub divisions are always by 60 at each step because it is a simple factor of 360 and there were 6 days weak (sadaha) in vedas. 1 extra day was added to some weeks making it 7 days, this day was not regular weak day hence tradition of weekly holiday arose. Since moon's node makes 12 rounds when Sun makes 1 round, the clock also copies that motion. Minute hand makes 12 rounds when hour hand makes 1 round. Angular and time measurement both are divided into 60 units so that they tally with sun's motion. One rotation = 12 rasi = 360 Arhsa (degree) 1 Arhsa = 60 Kala (minute) 1 Kala = 60 vikala (second) Tatpara and paratpara are further divisions. Thus angular motion of sun corresponds to time units 1 rasi = 1 month, 1° = 1 day 1' = 1 danda, 1" = 1 pala 1 tatpara = 1 vipala etc. This division of angle continues throughout the world till day. Time units are slightly different, but still in divisions of 60. Actually hour is derived from 'Hora' (Ahoratra) i.e. two divisions of a rasi (like day-night divisions of a day). Earth rotates 1 circle i.e. 24 hours in 1 day, hence 1 hora in 1 hour. 38 Siddhanta Darpana 6. References to introduction (1) For units and dimensions any standard text book of physics for +2 or graduate standard may be referred. First chapter is on units and dimensions. (2) For vedic astronomy see P.V. Holay's book. (3) For other interpratation of vedas see - Issues in vedic Astronomy and Astrology published in 1992 by Rashtriya Veda Vidya Pratishthan, New Delhi-2. (4) For details of siddhanta texts individual texts may be referred. Some information is com- piled in chapter 7 of Indian Astronomy - A source book. Translation of the text (Chapter I) Verses 1-9 - Mangalacarana - Prayer to Lord Jagannatha and other gods. Verses 10-11 - Pratijna - Scope and purpose of the book - It deals with only ganita jyotisa. Purpose is to explain difficult methods of mathe- matics in simple language to a common man. Verse 12 - Mathematical methods (Patiganita) has already been perfectly explained by Sri Bhaskaracarya in his text Lflavaa* . Without repeat- ing the same, motion of planets is discussed straight away. Verse 13 - Comparison with earlier Siddhantas - Some special subjects have been dealt with in this siddhanta, not found in earlier texts, for satisfaction of the learned. Verses 14-15-Importance of jyotisa - Veda guides everyone in yajna. Muhurta (auspicious Measurement of Time 39 time), for that is known through jyotisa. If veda is taken in human form, Jyotisa is eyes, Vyakarana is mouth (grammer), Nirukta (dictionary) is ears, kalpa (Purana) is hands, Siksa Sastra (reading) is nose, chanda (prosody) is feet. Thus jyotis is the part of veda through which all other parts can be understood. Comments - Many portions of vedas, Samhitas and all the text books explain jyotisa in similar words which need not be repeated. Verses 16-19 - About Siddhanta - Without mathematical astronomy, the whole jyotisa is ussless. Siddhanta deals with time scale from Truti (smallest unit) to Kalpa (biggest unit used in jyotisa), arithmatics (including indeterminate equa- tions of first and second degree), algebra, evolution and creation of world, orbits of earth, planets and stars, eclipse and conjunction of planets and description of various instruments like jya, dhanu etc. and elements of mensuration. Among all sastras jyotisa is highest; in jyotisa itself, siddhanta is best; and in siddhanta also gola (= sphere including Bhugola = geography and khagola = astronomy) is most important. A country prospers due to presence of men well versed in gola. Otherwise, animal behaviour 'spreads. Siddhanta gives all the four results (Dharma, Artha, Kama and Moksa). So Surya (creator of Surya siddhanta) has kept it a secret to be given only to good and pious prson. Verse 20 - First half of this text deals with time units, ahargana (count of days), bhagana (revolutions in sky), graha anayana (calculating planetary positions), jya (sine) kotijya (cosine) etc. 40 Siddhanta Darpana spasta sara (actual position of planets as seen in the sky), triprasna (three problems about daily motion). Second half deals with different theories, creation and dissolution (srsti and laya), earth (geography), kaksa (orbit), yantra (instruments), description of countries, prayers to lord Jagannatha and Kautuka Panjika (easy preparation of almanc). Verse 21 - Parabrahma was in the beginning. It created purusa and prakrti (2), mahattatva (intellect-1), ahankara (ego-1), tanmatra (elements 5), mahabhuta (5 types of creations or beings), 5 organs of sense, 5 organs of action, and one mind - a total of 25 elements. It supervises all. Verse 22 - Jyotisa Cakra - After creation, Brahma caused the sphere of akasa to rotate from east to west in a daily motion (seems as a result of rotation of earth on its axis from west to east) with respect to two dhruvas (north and south poles on the axis). With a slower motion, the planets move west to east relative to stars in nica and ucca circles (earth is not the centre of their or bits). Verses 23-30 - Kala has two meanings - one is destroyer of world and the other is reckoning of time. Time units are of two types - Suksma or amiirtta is very small unit which cannot be felt by senses, but calculated or measured by instruments. Prana is of 4 seconds, in which a person breathes in and out. This is the smallest sthula or murta time unit to be felt by senses. Divisions of time are - 1 lava = 100 truti 30 lava =1 nimesa = 1/135 seconds 18 nimesa = 1 kastha Measurement of Time 41 27 nimesa ■ Time to pronounce long vowel =0.2 seconds 20 long vowels =1 prana (4 seconds) 2 Pranas =1 Kala 3 Kala 1 vighatika or pala (24 seconds) 10 vighafi = 1 ksana 6 ksana = 1 ghati or danda (24 minutes) 2 danda 1 muhurta * 30 muhurta = 1 naksatra dina Note - Naksatra dina is the time of rotation of earth with respect of stars (23 hours 56 minutes) called sidereal day and is slightly smaller than civil day or solar day (between sunrise to sunrise) of average 24 hours. 30 naksatra day = 1 naksatra masa From sunrise to next sunrise is called savana dina (or civil day) 30 Savana dina = 1 savana masa 30 tithis = 1 candramasa NB-1 Candramasa is period from moon's node (amavasya - in same direction as sun, or Purnima, at 180° from Sun) to the same node next time. Tithi is l/30th part of that period equal to the time in which moon gains 12° difference over sun. Verse 31 - Detailed description of these naksatra and civil days, months etc. will be done in second part of the book. 12 solar months = 1 solar year Solar (Saura) year is one day of deva or asura (divya or Asura varsa) Note - 1. Sauradina is the interval of time during which sun moves 1° of ecliptic. In saura 42 Siddhanta Darpana masa it moves 30° or 1 rasi and in 1 year it makes a complete round of 360°. 2. Krantivrtta or ecliptic is the apparent path of stars from east to west in plane of equator. One complete round is called bhagana which is divided into 360° amsa (degrees). Each subdivision is in 60's as follows - 60 viliptas, vikalas, or seconds = 1 lipta, kala or minutes 60 liptas = 1 amsa, Bhaga or degree 30 amsa = 1 rasi or sign 12 rasis = 1 Bhagana or revolution Verse 32 - When sun is moving north of ecliptic (for six months), it is day for devas and night for asuras (day in north pole and night in south pole). When sun is moving south of ecliptic, it is night for devas and day for asuras. 360 divya or asura days = 1 divya or asuravarsa Verse 33-39 - Time scales greater than a year Period Divya years Solar Years Satya yuga 4,800 17,28,000 Treta yuga 3,600 12,96,000 Dvapara yuga 2,400 8,64,000 Kali yuga 1,200 4,32,000 Total - yuga 12000 43,20,000 or Mahayuga There is a sandhya, l/12th of the yuga, included in each yuga at its beginning and the end. Total sandhya is 1/6 of the yuga. Measurement of Time 43 Sandhya at beginning Divya years Solar years or end Satya yuga 400 1,44,000 Treta yuga 300 1,08,000 Dvapara yuga 200 72,000 Kali yuga 100 36,000 1 day of Brahma is called kalpa and it consists of 1,000 yuga. Kalpa is divided into 14 manvantaras of 71 yuga each. They are separated by 15 sandhya periods in between and at the end, in addition to manu period. Each sandhya is equal to one satya yuga i.e. 4/10 of a yuga. Thus, 1 kalpa = 14 manu + 15 sandhya = 14 x 71 yuga + 15 x 4/10 yuga = 994 yuga + 6 yuga = 1000 yuga Note - According to Aryabhata, a kalpa has 1008 yuga divided into 14 manus of 72 yugas each. Verse 40-46 - Current time - At present 50 years of Brahma have passed. In the 51st year (of 2nd parardha or half life of Brahma), this is the first day, called Svetavaraha kalpa. In this kalpa, six manvantaras have passed, namely - (1) Svayambhuva (2) Svarocisa (3) Auttami (4) Tamasa (5) Raivata and (6) Caksusa Current manvantara is Vaivasvata in which 27 yugas have passed. In 28th yuga, Satya yuga, Treta and Dvapara have gone. First fourth part of kali era is continuing. Time from beginning of creation till beginning of kaliyuga in this kalpa - Beginning sandhya " = 17,28,000 years 6 Manvantaras = 6x71x43,20,000 = 1,84,03,^0,000 years 44 Siddhdnta Darpana 6 sandhyas of 6 manus 6x17,28,000 = 1,03,68,000 years 27 yugas in 7th manu 27x43,20,000 = 11,66,40,000 Years Satya, Treta, Dvapara = 38,88,000 years Total = 1,97,29,44,000 years In kaliyuga, as in March 1996, 5098 years have passed. These are to be added to find the years passed since beginning of kalpa. Verse 47-52 - Motion of planets started at mid night at Lanka (a point of equator through which prime meridian through Ujjain passes). The day was named as Ravivara, Caitra Sukla pratipada. All the planets reach the same position at midnight of Brahma (after interval of 1 kalpa) At the end of Brahma's day (kalpa) all planets vanish. Author doesn't agree with Bhaskara that earth remains. Verse 53 - Bhagana is one revolution of a planet starting from Asvini to Revati end as seen in the sky. Explaination - Path of revolution along Zodiac (apparent path of planetary movement, more correctly of sun - rasivrtta) covers 360° divided in 12 rasis of 30° each. Almost the same circle is path of moon (inclined at 5° angle) There are 27 naksatras of 13° 20' each in which moon stays for about 1 day each. There was system of unequal division of naksatras also which will be discussed later on. Asvini naksatra is 1st and mesa rasi also start from 0° of zodiac. Last nakstra is Revati Verse 54 - Division of angular measurements according to previous acaryas - 1 Bhagana 12 rasi, 1 rasi = 30 Amsa Measurement of Time 45 1 Amsa = 60 kala, 1 kala = 60 vikala 1 vikala = 60 para 1 para = 60 vipara Note - Para and vipara are not used by other texts nor in modern mathematics. Verse 55 - Prayer of Lord Jagannatha Verse 56 - Sri Candrasekhara has written this drksiddha ganita (as observed in sky) in simple language, so that even children can follow it. Units of Time and Length -written and added later Units of measurement-A physical quantity is measured by 2 components- A basic unit of quantity used as standard for comparison of other quantities. A number which is ratio of measured quantity to standard .Basic unit should be available, reproduci- ble, convenient to handle and easy to compare through experiments. How many units are sufficient-For mechanics, 3 units are sufficient-length, mass, time. All other units of measurement can be derived from them. In 1901, Giorgi proved that that by adding a quan- tity related to electric properties, all physical quantities can be measured. To show the inter-relation between electricity and magnetism, or to explain the property of medium (vacuum or material), an- other quantity is required. Thus 5 quantities are sufficient to explain all physical quantities. These are called 5 Tanmatras in Sankhyg philosophy. For this 5 dimensional view of world, there are 5x5 = 25 elements in Sankhya. 5 fold division of units is described by 5 Ma (=to measure) chhandas- Ma, pra-ma, prati-ma, upa-ma, sa-ma For same type of quantity e.g. length- Ma- basic unit (e.g. meter), Pra-ma = multiples (kilometer etc.), Prati-ma - sub-multiples, Upa- ma = Related length units (Foot, Nautical mile, light-year etc.), Sa-ma = Link with other units (with time through velocity of light, or with area, volume etc.) Inter-relation of units of same or different kinds is called asn-vaya (upama + sama) Various types of inter-relations are called Vaya-chhandas. In context of length, they are classified as -.Ma =Prthvf (earth) -Standard rod, earth or earth-like compact body-sun, solar system, galaxy. Prama- Antariksa- Intermediate. Regions between and beyond earth(s). Pratima -Space, volume. /4s/7i/ADirections. Modern units of length-(1) Foot- Based on human foot- (2) Meter-It was defined in 4 ways-(i) Length of pendulum with half time-period of 1 second, (ii) 10 7 part of arc length from equator to north pole, (iii) 16,50,763.73 times wavelength of Kr 8 e radiation between energy states of 2 P 10 and 5 d 5, (iv) Distance traveled by light in 29,97,92,458 part of 1 sec- ond. (3) Nautical mile-1 minute arc of equator. (4) Astronomical unit (AU)-Semi-major axis of earth orbit around sun. (5) Persec -Distance at which AU subtends an angle of 1 second. (6) Light year-Distance traveled by light in 1 year=1016 meters approx. 1 Persec=3.26 light years, 1 AU=1.5x1011 meters. Seven yojanas-{\) Nara yojana =32,000 hands-used for human possessions of land. (2) BhO-yojana — 1000 or 1600 parts of earth diameter (surya-siddhanta -8km-yq/ana) (3) Bha-Yojana — 27 bhu-yojanas =216 kms, used for sun distance and size of galaxy (4) Prakasa-yojana — Distance traveled by light in 1 truti- 1/33,750 seconds (5) Dhama-yojana-Ksara dhama =720 parts of equator circumference = 55.5 kms. Aksara dhama-Measure of space with earth as standard in exponential scale. Number of powers of 2 is equal to akshara in chhanda. Distance d = r x 2 ( n - 3 ), r = radius of earth, n = unit of distance (6) Sun diameter as yojana for solar system in puranas. (7) Pramana-yojana — Starting from solar system, scale for each successive /oka in 500 times longer units at each step. Micro-units-Smaller worlds are successively 1 lakh times smaller-man (meter size), kalila (cell, 1 lakh part of meter), JTva (atom of 10(- 1 °) meter size), KundalinT (nucleus of 10 15 meter size), Jagat- particles of 3 types (10 20 meter size-not defined- Chara (lepton), sthanu (Baryon), AnupOrvasah (link -particles or meson) Deva-danava (10 25 meters size)-Creation from 33 types of devas only, not from 99 types of asuras (danava), Created world is 1/4 th part of purusa. 3/4 th is field or dark matter. P/fe/^proto-type, Parents)-10- 30 meter size. Rsi (string-rass/in hind/) 10 35 meters size. Micro or smallest unit called paramanu by Varahamihira = 8 4 parts of angula = 4.5 micron. Snpati calls it trasarenu equal to 60 atoms. So, atom =1.2 x 10 7 cm. Lalita-vistara-Paramanu-raja = Angula x 7 10 = 0.6 x 1 8 cm Tiloya pannati— Trasarenu = Angula x 8 9 = 1 .4 x1 8 cm. Single object or Brahma is indicated by Angustha (Thumb). In Purusa-sukta, angula means 96 parts of human length, or earth, solar system earth, galaxy as per context. Measures of Solar system-Modern estimate (NASA 2002) is that Woort cloud is boundary of solar- system at distance of 50,000-1,00,000 AU from sun. Indian measures- Samvatsara is aditya (energy-field) of sun. Fields of galaxy and universe are called Varuna and Aryama. This is sphere of 1 light year radius with center at sun. 1,575 crores diameter in unit of sun-diameter = 2.310 light year.(called ratha of sun). Outer wheel diameter is 6,000 i.e. outer boundary of Kuiper-belt. Modern estimate is about 70,000 plutonic bodies of above 100 km size. Puranas tell 60,000 balakhilyas of Angustha size i.e. 96 parts of earth diameter =135 km approx. Surya-siddhanta calls it Naksatra-kaksa of sun at 60 AU. This has been called A/oka (dark) earth of 100 crore yojanas (8km). Loka (lighted) earth is of 50 crore yojana (8 km) diameter. This has 7 dvTpa /oceans of Priyavrata formed by motion of planets. Inner wheel of sun is of 3000 sun diameter i.e. up to Uranus orbit. Prac- tical or Indra zone is of 1000 sun diameter {sahasraksa, aksa = sun or eye) up to Saturn orbit. Saturn being at the end of solar effect is called son of sun. Planetary distances in Bhuvar-loka are in terms of earth diameter. Size of next Svar-loka will be in 500 times bigger unit. Distance from sun to pole {Dhruva) is 14 lakh x 500 earth diameter which is distance of Woort cloud. Earth diameter x 2( 3 °) in aksara-dhama units. With three spherical zones inside earth as image of 2 bigger earths, there are 33 zones. Energy (prana) of each is a deva. Signs of 33 devas are letters from k to h - its arrangement (nagara) is Indian scripts called Devanagarl Earth/man = Solar earth/Earth. =10 7 , called koti (limit). This is called Maitreya- mandala or SavitrT (2 24 of earth size) in which creation occurs. Dyu (sky) of solar system is 10 7 times sun size. Earth is crore times or 24 dhamas bigger than man and is called GayatrT SavitrT x 2 24 = SarasvatT (creative field of galaxy). SarasvatTx 2 24 = Veda (creative field of universe or Veda- purusa^O times bigger than universe). Creative aspect is Niyati. Measures of Galaxy-(1) SOrya-siddhanta gives 1.87 x 10 16 Bha-yojanas (216 km) = 1,23,000 light years (modern estimate 1 lakh LY) (2) Earth size x 2 46 is galaxy Its creative field is KOrma (Go/oka of Brahmavaivarta purana) of 52 dhama units i.e.2 49 earth size. For 49 ahargana (dhama units) there are 49 letters in Devanagarl script from a to h. 3 extra units of KOrma is creator, conscious being is called ksetrajna in GTta chap- ter 13, so 3 letters are added at end-ksa, trajha. (3) Circumference is 0.5 Para (10™) dhama yojanas (55.5 kms). Diameter comes to about 1 lakh LY. (4) Size of KOrma in Narapati-Jayacharya\s hundred thousand (10 5 ) Sanku^O™) = 10 18 yojanas. As a purusaof Galaxy, it is 10 times bigger, so galaxy is about 10 17 yojanas. In space, earth is lotus of 1000 petals, 1 petal = 1 yojana (Aryabhata) (5) Galaxy is 1 crore times solar earth- SavitrT. (6) This is Janah loka of 2 crore yojana radius C\ yojana = 500 x 500 sun diameter) Bhuvar-loka is sphere of 15 ahargana {dhama) around earth i.e. 2 12 of earth size which is also called Varaha. Viewed from sun, it is 100 yojana ( = sun diameter ) high and 10 yojana high, up to lunar orbit. This covers up to 60% distance of Venus orbit. Earth's exclusive zone extends up to 9 ahargana i.e. 64 times earth radius. Moon is within it at 61 radius. Spiral arm of galaxy is called Sesa-naga. Near sun, it has 1000 sun like stars called 1000 heads of Sesa. Region of this is called Maharloka, whose size is given as- 1000 times size of solar system (sahasra-sTrsa purusa,\ 000 heads of Sesa) 43 ahargana =earth x 2 40 (tristup chhanda has 44 ± 2 or 43 letters in Mahesvara sOtra) 1 crore yojanas (1 yojana =500 sun diameters) Middle loka between earth (Jbhu) and satya loka 10 20 times bigger, so it is 10 10 times earth size. Tapah loka is given in 4 ways- 2 64 times earth size for 64 letters in BrahmJ script. 8 crore yojana (1 yojana =sun diameter x 500 x 500) (3) 864 crore light year radius -equal to day-night of Brahma. (4) Earth orbit/earth = Tapah loka/ga\axy Satya loka (1) Galaxy x 10 7 (or 2 24 ) (2) Maharloka /earth = Satya loka / Mahar (3) 12 crore yojana (1 yojana = sun diameter x 500 3 ) Time units Modern unit-1 second = 86,400 parts of mean solar day. Due to fluctuations and slowing down of earth rotation by tidal friction, new definition was adopted in1967-it is 9,19,26,31,770 times the period of light radiated by transition between two ground states of Cesium-133 atom. Nine Indian Time-units- (1)Brahma-T\me period of creation from formless {avyakta) to forms is called a day of Brahma or Kalpa. In same period of time, creation dissolves into avyakta {GTta 8/17,18). Day of Brahma has been defined as of 1000 yugas in GTta, puranas, Surya-siddhanta etc. Each yuga is of 12,000 divya-years, 1 divya-year =360 solar years. Thus, 1 day of Brahma = 432 crore years. (2) Prajaptya-Prajapati started yajha {GTta 3/10), so prajapatya period is period of galaxy from where creation started. Rotation period of galaxy is called Manvantara of about 31.68 crore years. Present stage is 7 th manvantara. This has been called 7 th day in Bible. After 4, 5 th days sun, moon, earth were created. So, day cannot mean here rotation of earth or even of sun. This is rotation of first creation galaxy. Modern estimate of period of sun revolution around center of galaxy is 20-25 crore years. (3)Divya year is of 360 solar years, arrived in 3 ways. This is approximate period of revolution of imaginary planet at 60 AU (or average rotation of about 60,000 Balakhilyas at same dis- tance), called A/oka (dark) boundary of 100 crore yojana diameter, This has been called pari- varta yuga in Vayu, Matsya puranas, which is cycle of historic changes. Third view is that north-south motion of sun is like day-night cycle. This cycle of 1 year is 1day (d/vya). Taking round number 360 for days in a year, d/vya year is of 360 years. (4) Guru scale-In period of 60 years Saturn and Jupiter complete integral revolutions-2 and 5. Alternatively, Ang/ra-effeci (upward convection due to radiation pressure takes 60 years to complete {Aitareya Brahmana 18/3/17, Taittinya Brahmana, 2/2/3/5-6) (5) Pitar mana- Synodic revolution of moon in 29.5 days is called 1day of pitars. Varaha is 15 ahargana or 4,096 times earth size-that is parjanya. Intermediate level is pitar, 64 times earth size called pitar. Moon orbit is 61 times earth size. So pitars of human beings also reside on outer region of moon. Our bright half of month is night of pitars and dark half is their day. (6) Savana mana-Sunnse to next sunrise is savana or civil (practical) day.1 month = 30 days, 1 year = 12 months. (7) Solar-Apparent revolution of sun around earth is yea r(sidereal). 1/12 th part is 1 month (1 rasi = 30°. Movement). 1 day is 1° movement. (8) Lunar-Synodic revolution of moon around earth is a lunar month. 5 angas of panchanga are defined from moon and sun — Tithi- (M-S) /12°, \tithi- 2 karana, yoga = (M+S) /13.3. Naksatra is average daily motion of moon. Vara (day) is cyclic naming of days. (9) Nakstra (sidereal)-Axial rotation of earth with respect to fixed stars is sidereal day of 23 hours 56 minutes. Month, year have 30 and 360 days. Micro units of t\me-Satapatha Brahmana (12/3/2/1,5) divides mean day length of 12 hours successively by 15 parts into units named as muhurtta (48 minutes), ksipra, etarhi, idanT, prana, aktana (or, ana), nimesa, lomagartta, svedayana. It is stated that stars (naksatra) in galaxy (Brahmanda) are like its loma-gartta (roots of skin hairs). Number of lomagartta in a year (Samvatsara) is equal to the number of stars in a galaxy, so this unit of time is called lomagartta which is equal to muhurtta x 15 7 = about 80.000 parts of a second. Similarly its 15 th part is svedayana equal to about 11, 20,000 parts of a second. In this time, light travels about 270 meters. Rain drops (sveda) move (ayana) about same distance without breaking or joining, so this time unit is called svedayana. Estimate of number of stars in galaxy was done after 1985 which is correctly estimated in Satapatha-Brahmana to be 10< 11 ). Logic of division by 15 is given that ratio of earth orbit to earth size is same as ratio of Tapah-loka (visible universe) and galaxy, both equal to 2 < 15 ). Seven Yugas-By joining two cycles of time, a yuga is formed. MunTsvara in his astronomy text Siddhanta Sarvabhauma has stated 5 yugas- 5 years, 5 x 12 = 60, 12 x 60 = 720, 600 x 720 =kaliyuga, kalix 10=1 yuga. Like 7 yojanas, there are 7 yugas, depending on completion of various yajhas — (1) Sanskara yuga — Education and other reforms projects are completed in 4 to 19 years which is a sanskara-yuga. (a) Gopada-yuga-\s of 4 years like modern leap year system. Its year starts in godhuli-vela (literally cow-dust- time, when cows return home at sunset, dust is raised) like Hebrew or Islamic calendars. West Asia was place of Asuras, called Nisacharas because their day started with sunset.This is described in Aitareya Brahmana (7/13). Suppose, 1 st year starts at 6 PM on 4-1 -2001. This year KaliW\\\ end at 12 PM on 4-1-2002 when people will be sleeping, So, kalixs called sleeping. 2 nd year dvaparaW\\\ end on 5-1-2003 at 6AM, so dvapara is called twilight. 3 rd year tretaW\\\ end at 12 AM on 5-1-2004, when people will be standing, so tretaxs called standing.4 th year will end on 4-1-2005 at 6PM as there is leap year in 2004. Here year is of 365 1 / days, (b) 5 year yuga - Yajusa jyotisa has described 5 years yuga in which lunar years match with solar year by adding 2 extra (adhika) months. Years are named by adding prefixes sam-, pari-, id-, ida-, anu-, to the word vatsara. (c) 12 years yuga is revolution period of Jupiter around sun. These are named like months of lunar year-chaitra, valsakha, etc. (d) Rahu yuga-\\ is called Saros cycle in Babylonian astronomy. This is relative motion of Sun and Rahu (node of Moon) in 18 years 10,5 days in which eclipse cycle repeats. Its half period of 3339 tithis is also approximately eclipse cycle stated by Visvamitra (Rk 3/9/9). (e) 19 year yuga — This is followed in Rk-jyotisa, as explained by Sri Prabhakara Ho/ay Nagpur. In this period, lunar years match with solar year more accurately (less than 2 hours error) by adding 7 adhika months. Years are classified into 5 types according to 5 blocks of 6 tithis in which a solar year starts. In 1 yuga, there are 5 years of samvatsara type. (2) Manusya (human) yuga - 60 years active life of man is called Angira period in which 6o year cycle of guru-years occurs. In 100 years, Saptarsi (Ursa major, ursa -rsi) moves / naksatra, i.e.27 th part of zodiac circle. In Rk- jyotisa calculation, moon moves 1 naksatra ahead in 100 years. The line joining two eastern stars moves 1 nak- satra back in 100 years. This year count has been called Laukika in RajataranginT. 1/3 rd of divya-dina or pari- varta yuga of 360 years is 120 years which is human life for astrological timing of events. (3) Parivarta yuga-Th\s is divya-dina of 360 years in which historical changes (parivartana) occurs. 71 such yugas make manu-yuga of 26,000 years (precession period of earth's axis) in Brahmanda Purana (1/2/29/19) (4) Sahasra yuga-Bhagavata purana (1/1/4) states 1000 year satraof Saunaka in Naimisaranya. Compilation of Pu- rasas took about 200 years, but its effect on social norms lasted for thousand years. It could be revised only in time of Vikramaditya of Ujjain (82BC-19AD, era started in 57 BC) as per Bhavisya purana (3/3/1/2-4). During 3000 years, seasons shifted back by 1 Vi months.720 years of MunTsvara is of 2 parivarta yugas. Sahasra (1080) years is 3 parivarta. Even Gautama Buddha planned his religion for 1000 years. Prophet Mohammed predicted Islam to last for 1400 years. Saptarsi yuga is of 2700 years. It is described in two ways in Brahmanda, Vayu puranas. 2700 solar years are called divya years, Manusa year is 12 revolutions of moon around earth in 327.5364 days. Saptarsi eta is also stated to be of 3030 manusa years =2717 solar years. Romaka yuga of Pahchasiddhantika (Varahamihira) is of 2850 years =19 year Rk-yugax 150, (5) Dhruva or Krauhcha yuga-Jh\s is of 9090 manusa years or 8100 solar years. This is exactly 3 times saptarsi yuga and about 1/3 rd of Ayana or Manu-yuga. Position of north pole of earth makes a circle in 26000 years and is close to 3 stars so the period is divided into 3. On earth regions around north pole are called Krauhcha-dvTpa, so it is called Krauhcha yuga also. In north India, Guru years are calculated as per actual mean motion of guru in 361.14 days (Surya-siddhanta). In 85 solar years there are 86 guru years. In south India, solar years are named as guru years (Paitamaha-siddhanta). In 85 x 60 = 5100 years, both cycles are completed. On 11-2- 4433 BC when Rama was born, the year was start of guru cycle in both systems (1 st Prabhava year) as per Visnudharmottara purana (82/7,8). Matsya incarnation had occurred 5100 years before that in 9533 BC. He- rodotus gives date of sinking of last island of Atlantis in 9564 BC. This is approximately period of last glacial flooding. (6) Ayana (Precession) yuga-Earth's axis rotates around pole of ecliptic (earth orbit) in 26, 000 years. This has been called Manu yuga in Brahmanda purana (1/2/29/1 9). Glacial age on earth is due to two cycles-Precession in 26000 years in reverse direction and advance of earth aphelion in 1 lakh years. Glacial region is around north pole. When it is inclined away from sun or earth is at aphelion it gets less heat. When both combine, it is Glacial ice age. Its cycle is in 21600 years— 1/21600 = 1/26000 +1/100000 Civil cycle is taken as middle of the two. By taking 24000 years cycle, there is positive error in 12000 years and negative in other half (Brahma-sphuta-siddhanta, madhyamadhikara, 60, Siddhanta-siromani of Bhaskara-ll, Bhu-paridhi, 7). Each half of 12000 years is taken as yuga of 12,000 divya years in every purana, Mahabharata, etc. In Mahabharata period AvasarpinF (descending) period was running in which part yugas -Satya, treta, dva- para, kali, of 4, 3, 2,1, parts come in that order. Of this order, /(a// started on1 7/1 8-2-31 02 BC Ujjain mid-night. The other half is called UtsarpinT (ascen<S\ng) in which kali to satya yugas come. (7) Astronomical yuga — This is 360 times ayana yuga of 43,20,000 years in which all planets up to Saturn complete integral number of revolutions (Bhaganopapatti'm Siddhanta-Siromani of Bhaskara-W). Two other cycles depend on it which is not verified so far-(i) Movement of magnetic pole and magnetic reversal, (ii) Movement of geo- graphical pole in north south direction (Indra-Vijayaoi Madhusudan Ojha) or equivalent continental-shift. Time has been equated to full-pot (Purna-kumbha) or volume .and parallel with 7 chhandas is shown in Kala-sukta of Atharva-veda (19/7), BrhatT-sahasra (36000) days is life period of man (Aitareya Brahmana). Understanding of these yugas explains puranic chronology since 62,000 BC. (Detail article with references is in Hindi, titled — " Chhanda-Adharita Mapa-Vijnana" Chapter - 2 REVOLUTION OF PLANETS Subject - This chapter deals with total revolutions of planets in a kalpa, adhika masa (gain of lunar months above 12 in a solar year), ksaya tithi (shortage of lunar dates from months of 30 civil days), rotation of orbits. From that; the average daily motion of planets have been calculated. 1. Explanations - All the results given in this chapter are assumed and no hint is given as to how these numbers have been found. Obviously they are highly accurate and have been followed since time immemorial. All texts from Suryasiddhanta to Siddhanta Darpana have fol- lowed the same practice. It is possible that the samhita and Brahmana texts gave the methods and observations of planetary motions. The teaching of science and mathematics was like present day text books -of college, and not in verse form which is useful for memorising only. This became necessary when educational institutes and their books were destroyed due to foreign invasions. Mathematics in modern text book form has been found in Bakhsali manuscripts of mediaval preiod (edited in 3 vols by G.R. Kay - New Delhi-7) All ancient authorities have admitted that these results are not based on observations. Surya siddhanta has stated that these were given by Surya to Mayasura in Romaka town in 21, 63, 223 B.C. (121 years before the end of Satya yuga). To some extent, it is correct. Even with most modern Revolution of Planets 47 equipments, calculation of motion for billions of years cannot be made on observations during a life time only. It needs systematic observations for at least 500 years for studying motion in orbits, and at least 10,000 years and much more, if rotation of orbits, or change of earth's axis is to be calculated. Thus the result could have been otained only from observations through the ages, preserved by generations (like Vedas, it has to be 'Apauruseya' i.e. god given or beyond a human being). 2. Origin of complete revolution numbers in a kalpa - There are two assumptions by ancient authorities - In general, it is assumed that the figures have been obtained on the basis of observations through ages. Total motion in a kalpa has been calculated on basis of observed rates. Siddhanta Darpana has followed pattern of Siddhanta Siromani of Bhaskara II except for some new improvements. Bhaskara has assumed that concept of yuga and kalpa has been derived from the observed motion of planets. The planets repeat their positions after every yuga (the grand year), as in civil year earth comes back to the same position round the sun. However, if we consider rotation of orbits; its cycle is repeated only after 1000 yugas or a kalpa. For example, orbit of Saturn rotates only 39 times in a kalpa, so its motion cannot be perceived within a yuga. Text books of Tantra and karana are not concerned with such slow motion. Another presumption is that theories of planetary motion and constants of orbit have been given in vedas. We do not know the technical terms and method of presentation of astronomy as explained in samhita and Brahmana texts. Varahamihira was probably last who * understood 48 Siddhanta Darpana contents of all 3 parts of jyotisa from vedas. It is presumed that 10,000 verses of Rkveda contain records of astronomical observations for 10,000 years or yugas of 5 or 19 years. Though only Aryabhata I has specifically mentioned two motions of -earth, it appears that many others knew about movement of earth. It is clear from names Jagat (moving), samsara etc. Methods of calculation followed by other astronomers also indicate that they were following some theories known in vedas or other texts but not specified in astronomy works. Whatever may be the nature of planetary motion, it contines to be observed against the background of same Zodiac of 12 rasis or 27 naksatras, and from earth only. From scattered observation charts through the ages, theories of circular or elliptical orbits have originated. They cannot be observed directly. 3. Circle and Ellipse Circumference 6 Centre, radius OP or OQ Fig. 1A-Circle AB = Major axis, CD = Minor axis O Centre, Focus F1 and F2 Fig. IB-Ellipse Definition - Circle is path (locus) of a point (circumference) which remains at fixed distance (radius) from a fixed point called centre. (Figure 1 A) Revolution of Planets 49 Circle is a round figure on a paper looking same from all directions. Ellipse is elongated form of circle - stretched in two opposite directions (called major axis) Definition - Ellipse is locus of a point whose sum of distences from two points Fi and F2 (called focus) is constant. Thus DF1+DF2 = D'F 1 +D'F 2 = AB (major axis). Smallest width is minor axis CD. AB and CD are perpendicular at their middle point O, called centre (Figure IB) Kaplar's laws of planetary motion indicate that planets move in an ellipse round the Sun which remains at one of the focus (not at the centre). Newton's law of gravitation were derived from Keplar's laws and vice versa. In circular orbit (special type of ellipse where both focus are at same point), speed of planet will remain constant. In elliptical orbit, it will be fastest when the planet is closest to Sun (at A if sun is at Fi). It will be slowest when farthest from sun (at B). B is called Aphelion (Apex=top, helios = Sun) or mandocca (slow+top) or ucca in short in jyotisa. One feature of circular orbits remain the same in elliptical orbit al$o. Though the speeds vary at different position, the area covered by line from sun to the planet in unit time remains the same. Thus POQ area in circle or DFiD' area in ellipse, covered in unit time remain constant. 4. Sighrocca and Mandocca - The relative motion of sun and earth remains the same, whether it is observed from sun or earth. In either case, it will be elliptical motion with same speed. Similarly, 50 Siddhdnta Darpana moon also moves in elliptical orbit round the earth. The position of sun or moon where its speed is lowest (at highest point in orbit), is called man- docca. Orbits of other planets around sun are also elliptical. However, when we observe from earth, it is a composition of two elliptical motions - one is relative motion of sun around earth and the second is motion of planet round the sun. The planet in smaller orbit is called sighrocca, as the average motion is faster in smaller orbit. The highest point in slower and bigger elliptical orbit is called mandocca. In bigger orbit, a planef s motion will appear slow due to two reasons. At lafger distance, gravitational attraction of Sun is small and planet moves at small speed to counter the attraction by its centrifugal force. If speed is more, it will go still farther and loose speed, till it settles into a stable orbit. Due to larger distance the angular speed appears still slower. 5. Pata- Orbit of sun around earth and orbit SS' = major axis of sun's orbit MM' = Major axis of Moon's orbit R, K = Point of inter- section of two orbits of moon around the earth are hot in the same plane. They are inclined at angle of about 5° (Figure 2). Revolution of Planets 52 If sun orbit is taken as reference level, at point R, moon appears moving towards north or upwards. Moon itself is not at R, it may be at any point on its orbit MKM'R. R is merely an imaginary point of intersection and is called uttara-pata (ascending mode) or Rahu. At point K motion of the moon appears south wards, hence it is called daksina-pata (descending node) or ketu. When moon is at one of the patas on purnima (180° away from sun) or amavasya (same direction as sun), eclipse occurs. Thus Rahu and Ketu are said to cause eclipse. Rahu and Ketu are called chaya graha as they are only imaginary points. They have nothing to do with shadow of earth or moon. R and K are always in opposite direction from earth as seen from the diagram. They are moving in reverse direction to the direction of planetary motion. Their revolution is called bhagana of pata (in about 19 years). Similarly, pata of other planets also move. But their motion is so slow that it is noticed only in a kalpa. Motion of Ucca - Motion of ucca of moon is visible in a yuga (one revolution in about 9 years). Motion of other planets is very slow and can be noticed only in a kalpa. 6. Change in values of bhagana - Surya Siddhanta, first chapter states that motion of planets vary with time and hence its observation needs to be corrected after long lapse of time. It is known in modern astronomy that earth's rotation on its axis is slowing down at the rate of 52 Siddhantd Darpana 14 seconds per century due to tidal function. Due to decrease in angular momentum of earth, moon is moving away at the rate of 8mm every year to conserve the angular momentum of earth-moon system. Due to tidal forces of galaxy and sun and friction of solar atmosphere, motion of planets also will slow down. But its values are not known either in siddhanta texts or in modern astronomy. It can be inferred to some extent by comparison with old records of solar eclipse in vedas, or comparing old values of bhaganas with present values. Thus, if we calculate the average motion of the planets on the basis of their total motion, their values will differ from the real observation. Another reason of error will be inaccuracy and approxima- tion of mathematical methods and calculations. To correct these, various astronomers have introduced correction terms for their era. Candrasekhara was last among them. In addition, he introduced 3 correction terms for moon's motion, whose error was noticed due to its faster motion. REFERENCES (1) For knowledge of circle and ellipse any college text book on plane coordinate geometry can be referred. For example Loney's coordinate geometry. (2) For concept of intersection of two orbital planes any book on solid geometry can be referred. Fuller discussion will be in books of spherical Trigonometry by Gorakha Prasada or by Todhunter. Revolution of Planets 53 (3) Development of planetary theories of motion have been excellantly explained in 'The struc- ture of the Universe' by Sri J.V. Narllkara. (4) Historical discussion of zodiac and factional slowing of planetary orbits is given in 'An intelligent Man's guide to Science' by Isaac Asimov. (5) For comparison of values of bhaganas given in different texts any good commentary of standard texts may be referred. One may read the histories of astronomy, referred to earlier in introduction. Translation of the text Verses 1-2 - Bhaganas of planets in a kalpa (West to east) Ravi, Budha, Sukra 4,32,00,00,000 Candra 57,75,33,36,000 Mangala 2,29,68,71,112 Brhaspati 36,41,55,205 Sani 14,66,49,716 Budha SIghrocca 17,93,69,67,141 Sukra SIghrocca 7,02,22,57,860 Mangala, Brhaspati, Sani SIghrocca 4,32,00,00,000 Note : 1 - Sun - Ravi, Surya, Arka etc. in sanskrta Mercury - Budha; Venus - Sukra, Mars - Mangala, Kuja, Bhauma; Jupiter - Guru, Brhaspati; Saturn - Sani 54 Siddhanta Darpana 2. Budha, Sukra are in inner orbits around sun, so their revolutions are same as sun as they appear tied with it as seen from earth. Their revolution is equal to number of solar years in a kalpa by definition (1 year corresponds to 1 revolution of sun) 3. Sighrocca of Brhaspati, Sani and Mangala is due to earth's orbit round the sun. Hence it is equal to apparent revolution of sun round the earth. Verse 3 - Mandocca Bhagana in a kalpa from west to east. Siddhanta Darpana Surya Siddhanta Ravi 334 387 Candra 48,81,17,940 48,82,03,000 Mangala 310 204 Budha 410 368 Guru 805 900 Sukra 557 535 Sani 70 39 Note - Only source of these figures is Surya siddhanta. Author has not indicated source of his corrections. Verse 4 - Bhagana of pata (East to West) Note - Pata is calculated according to inclina- tion of orbit with Ecliptic. Since it is path of sun, there is no pata for sun. Planets Bhagana in a kalpa Surya-siddhanta Candra 23,22,98,033 23,22,38,000 Mangala 298 214 Budha 552 488 Guru 945 174 Revolution of Planets 55 Sukra 110 903 Sani 545 662 Note - Source of different figures and large variations in figures for guru and sukra is not explained. Verses 5-6 - Naksatra dina is the time between rising of any naksatra to its next rising (equal to time of revolution of earth on its own axis) The time between rising of a planet to its next rising is called savana dina for that planet. (For example sunrise to next sunrise is savana surya dina). This corresponds to rotation of earth with respect to that planet, (i) Total number of naksatra dina in a kalpa 15,82,23,78,28,000 (ii) Savana dina of a planet = Naksatra dina - graha bhagana (iii) Candra masa = Candra bhagana - Surya bhagana Verse 7 : In a kalpa (or a Mahayuga) No. of adhimasa = No. of candramasa - No. of Sauramasa No. of Kshaya dina (Lost days) = 30x No. of Candramasa - No. of savana days Verses 8-11 : No. of solar months in a kalpa 51 ,84, 00,00, 000 No. of Candra months " 53,43,33,36,000 No. of adhimasa " 1,59,33,36,000 No. of Sauradina " 15,55,20,00,00,000 No. of Candra dina " 16,03,00,00,80,000 56 Siddhanta Darpana No, of Surya (savana) dina " 15,77,91,78,28,000 No. of Ksaya tithi 25,08,22,52,000 Note - No. of savana dina and ksaya tithi here is same as that of Surya siddhanta, where the figures given are for a mahayuga. However, savana days in a mahayuga are different according to other texts - Surya siddhanta of Panca siddhantika - 1,57,70,17,800 Aryabhata - 1,57,79,17,500 Brahma sphuta siddhanta, siddhanta Siromani ' 1,57,79,16,450 Mahasiddhanta - 1,57,79,17,542 Verse 12 - Definition - At any given time kendra of a graha (angle) = position of a planet- position of its ucca. Compared to Sighra ucca it is called SIghra kendra, compared to manda ucca, it is called manda kendra. Ucca and pata bhagaras are not completed in a yuga except for moon, so their bhaganas are stated for a kalpa (1,000 yugas) Verse 13 : Sighrocca = drak, cala, asu, capala etc. Mandocca = Mrdu, ucca, manda etc. (synonyms) Verse 14-15 - No. of asu (prana = 4 seconds) in a day- 1 average (madhyama) naksatradina = 21,600 asu 1 madhyam saura dina = 21,976 asu 1 madhyam Candra dina = 21,320 asu 1 madhyam savana dina = 21,659 asu Revolution of Planets 57 Savana dina is commonly used by people which is divided into 60 ghatika or danda. Verse 16 - Bhagana = 1 complete revolution = 360° amsa. Bhagana kala = bhagana x 360 x 60 Dainika kala of a graha = graha bhagana kala in kalpa/savana dina in kalpa Time for 1 bhagana of a graha = savana days in kalpa/graha bhagana in kalpa Verses 17-18 : Like division of full circle rota- tion in 360° (amsa) and then further sub-divisions by 60 in each step, learned men have divided a savana dina also by 60 at each step to danda and pala etc. One complete revolution (bhagana) of sun takes days 365/15/31/31/24 danda, pala etc. Verse 19 - Madhyama guru takes days 361/5/27/27/13 in one rasi at average speed. Verses 20-24 - Daily motion of planets is described in liptas (1/60 amsa) and 10 furhter sub-divisions in steps of 60. By multiplying this daily motion with no. of days (passed from beginning of kalpa to desired day), madhyama graha (position with average speed) is obtained. Sun (Ravi) 59-8-10-10-24-12-30-4-10-4 Candra , 790-34-52-3-49-8-2-16-10-11 Mangala 31-26-30-6-47-44-32-49-3-4 Budha SIghra 243-32-16-7-17-17-59-43-42-44 Guru 4-59-5-37-0-36-41-17-1-51 . Sukra SIghra 96-7-37-47-57-50-39-32-31-35 Sard 2-0-26-55-2-53-21-2-4-54 Candra ucca 6-40-54-31-0-44-5-52-45-39 Candra pata 3-10-47-40-40-26-11-25-13-30 58 Siddhanta Darpana Verse 25-26 - Kranti vrtta in sky is the sudarsana cakra of Jagannatha with which he removes fear, produces light and destroys all in the end. With this prayer Sri Candrasekhara Simha completes second chapter of Siddhanta Darpana describing bhagana of grahas. Chapter - 3 MEAN PLANETS Scope - This chapter describes methods for calculating value of madhya graha (position calcu- lated from average motion). This coincides with sphutagraha (true position) twice in every bhagana (revolution). Since the planetary motions started from mandocca position; at mandocca, sphuta and madhyama positions should be same (for sun and moon). Verse 1 - Ahargana (count of days) - Ahargana for ista dina (desired day) is counting of days from beginning of kalpa (in siddhanta text). This is needed to know the graha on ista dina of any varsa, masa or tithi. Note - In tantra, ahargana is counted from beginning of mahayuga (or ^sometimes, from the beginning of kaliyuga). In karana text, ahargana is counted from any reference year or beginning of current year itself for preparation of panjika. Verses 2-8 - Steps in calculation of ahargana - 1. Add the saura varsas for 6 manu, 7 manu sandhi (each equal to satya yuga), 27 mahayuga, 3 padayuga and years passed in current kaliyuga. Note : In the present Svetavaraha kalpa, 6 manus out of 14 have passed. In the current 7th vaivasvata manu, 27 yugas have passed. At beginning of kalpa and after each manu, a sandhi equal to one satya yuga exists! In current 60 Siddhanta Darpana mahayuga, satya, Treta and Dwapara have passed. Kaliyuga started on 17/18-2-3102 B.C. Ujjain mid- night. 2. Deduct 1, 70, 60, 400 years Note - According to verse 24 of madhyamadhikara in Surya Siddhanta, Brahma took this time of 47,400 divya varsa to create stars, planets and living beings. The present stable motion of planets started after that. 3. Multiply by 12 to make it months and add the number of months (masa) elapsed from Caitra (Candra months in current year are almost equal to saura masa). 4. Keep the result (no. of completed saura months) at two places. 5. At first place, multiply it by no. of adhika masa (1,59,33,36,000) in a kalpa and then divide it by sauramasa in a kalpa. Result will be adhimasa related to the saura varsa. 6. Add this to the no. of masa from kalpa beginning obtained at step 3. 7. Multiply Candramasa by 30 and add the days completed in the present month (Candra masa) 8. Keep the result at two places. 9. At one place, multiply it by kalpa tithi ksaya (25, 08, 22, 52, 000) and divide by number of kalpa tithi. Substract the result from kalpa tithi at the second place. Difference is number of savana tithis from kalpa beginning. Divide it by 7. Remainder will give the week day counted from ravivara (sunday) as 1. Mean Planets 61 Mathematical comments - 1. The methods are based on rule of 3 (Trairasika) or ratio and proportion. , v Adhimasa till ista dina Adhimasa in a kalpa /^\ U _ k. Sauramasa till ista dina Sauramasa in a kalpa ... ksaya tithi till ista tithi ksaya tithi in a kalpa ^ ' gata tithi (elapsed tithi) Total tithi in a kalpa Tithi is a candra dina. 2. Ratio between Candra and saura masa, tithis; Saura masa + adhimasa = Candra masa Within current year, they are almost equal. Candra masa x 30 = Candra tithi Candra tithi is almost equal to savana dina in a current month Candra tithi - ksaya tithi = savana dina. Savana dina is time from sunrise till next sun rise. 3. Kalpa had started on ravivara at midnight at Lanka which is at equator on 0° longitude of India (passing through Ujjain). Verses 9-13 : Errors in approximation of sauramasa and Candra tithis (as explained in mathematical notes above si 2) Adhimasa - While calculating adhimasa only the quotient (result) is taken and remainder is left out. If remainder is almost equal to divisor, or if an adhimasa has passed recently (in past 1 year), then 1 is added to know correct adhimasa. However, if the remainder is almost zero or an adhimasa is to come soon, then 1 is to be substracted. Ksaya tithi - Similarly, if in calculation of ksaya tithi, remainder is almost equal to divisor 62 Siddhanta Darpana and within a week ksaya tithi has passed, then 1 is added to the result. (If pancami comes after tritiya, then caturthi is ksaya tithi upto dasaml, if remainder is more than half the divisor, 1 is to be added to ksaya tithi. Thus 1 will be substracted from ahargana. If remainder is almost zero, 1 is added to ahargana. Correctness of ahargana can be checked with week day. Verse 14 - Masadhipati - Divide ahargana by 30, multiply the result by 2, add 1 and divide by 7. Remainder will indicate week days counted from ravivara as 1. (soma 2, mangala 3, budha 4, guru 5, Sukra 6, Sani 0) Ruler of this day will be masadhipati. Derivation - Each civil month is of 30 days (civil). Ruler of 1st day is masadhipati. Ahargana divided by 30 gives the number of civil months. In each month of 30 days; 4 weeks are completed (4x7=28 days) and 2 days remain. Hence for each month; 2 remainder days are taken. 1 is added because the first day of kalpa was ravivara, 1st day. Verse 15 - Divide ahargana by 30. Remainder is the days gone (gata dina) in current month. Gata dina substracted from 30 gives bhogya (remaining days) dina of the month. Derivation is obvious from earlier verse. Verse 16 * Divide ahargana by 360, Multiply result by 3 and add 1. Divide the result by 7. Remainder indicates week days starting from ravi as 1, which is the varsapati. The remainder left after division of ahargana is bhukta dina (past days) of current year. Mean Planets 63 Comments : (1) Masasidhipati and varsadhipati are used only for calculating kala bala in horoscopes, or in mundane astrology for forcasting events of the year. It has no importance in ganita jyotisa. (2) Each civil year is of 360 civil days. Hence the quotient after division by 360 into ahargana, is number of completed civil years. Remainder will be past days of the current year. (3) In 1 year of 360 days, 360 -s- 7 = 51 weeks and 3 extra days remain. Hence each completed year gives 3 days for count of week days, Next day will be first day of current year, hence 1 is added to find varsadhipati. Verse 17-20 - Lord of first day of masa (month) is masadhipati, and lord of first day of varsa is varsadhipati. Satananda (author of Bhasvati karana) and his followers have different opinion. Lord of the day on which mesa samkranti falls is the lord of the year (varsadhipati). To calculate the number of days in that year, the following rule has been given. Calculate the danda, pala etc. from time of entry of ravi in mesa samkranti to the time of beginning of next day. Multiply it by 4 and keep it in 3 places. Divide the number at third place by 37 and add the result at second place. Divide at second place by 8 and add this result at first place. The result in danda etc. will indicate the number of days for which the varsapati will rule. For remaining days of the year, (360 - days of rule of varsapati) lord of day next to sankranM will rule. Siddhanta Darpana 64 According to this rule, no graha can rule for more than 271 days. Mathematical symbol : let T = time in danda etc. from entry of ravi in mesa to next sunrise. 4T 37 = T'+R (remainder smaller than 37) 4T + T _ j " + R' (remainder smaller than 8) 8 4T+T" = D danda + p pala etc. D is the number of days for which varsapati will rule. Lord of the day after mesa Sankranti will rule for 360 - D days. 2 This appears to be a convention by Satananda, hence no derivation of the rule is given. 3. Maximum days of rule of varsapati - T < 60 danda 4T 4X60 = fi5 T < 37 Hi7~ 4j + j' 4 X 60 4- 6.5 t t» < — g- < r^ D = 4T + T" < 4 x 60 + 30 = 271 Hence maximum days of rule from sankranti day is 271 days. Verses 21-22 - Formula for calculating graha for indicated day - Multiply ahargana by kalpa bhagana and divide by kalpa savana du£ ResuU wiU be lapsed bhagana. Multiply remainder by 12 and again divide by kalpa savana dina. Again multiply by 30, 60 and 60 and divide by kalpa Mean Planets 65 savana dina to obtain amsa (degree) kala (minutes),, vikala (seconds) Explaination (1) By ratio and proportion Bhagana till ista dina _ Ahargana Bhagana in a kalpa ~ Savana dina in a kalpa (2) Fraction of bhagana are converted to rasi etc. according to the scale - 1 Bhagana = 12 rasi, 1 rasi = 30 amsa 1 amsa = 60 kala, 1 Kala = 60 Vikala + (3) 1st rasi is mesa starting from 0° to 30° in kranti vrtta (ecliptic). 0° starts from a fixed point marked by star groups in Indian astronomy. In western system, 0° is marked by point of intersec- tion of equator with ecliptic plane, where- motion of sun appears northwards. Difference between the two initial points is called ayanamsa. Axis of earth rotates one round in 25,762 years. In Indian system also calculation of day length, lagna etc. are done from this ayanamsa sayana point. (4) 12 rasis are 1. mesa 2. vr$a, 3. mithuna, 4. karka, 5. simha, 6 kanya, 7. Kite, 8. vrscika, 9, dhanu, 10. makara 11. kumbha, and 14. mina. Verse 23 - (Quoted from Surya Siddhanta) - Same method is used for calculation of Slghrocca, mandocca and pata for ista dina. However, for pata, the result will be deducted from 12 rasi, because movement of pata is in opposite direction of graha. Note - When it is unnecessary to explain in more detail, the author has just referred to quotation from previous authorities - mainly surya sidhanta or siddhanta siromani. Sometimes quota- 66 Siddhdnta Darpana tions have been given for comparison or contradic- tion on important points. Verses 24-25 - Calculation of guru varsa - calculate bhagana of guru as before and add 3 (bhaganas) Multiply the sum by 12 and add their rasis lapsed and add 2 again. Divide this sum by 60 and add 1 to the remainder which indicates guru varsa counted from Prabhava etc. Notes: (1) Secret of guru varsa has been explained in chapter 21 of this book. (2) Guru takes about 12 years to move around sun and about 1 year to cover 1 rasi. Hence guru varsa (time in a rasi with medium speed) is similar to saura varsa (time of 12 rasis or complete bhagana) Guru varsa is called samvatsara of 361.02672 savana days which is smaller by 4.23203 days from saura varsa and bigger by 1.02672 days from savana varsa of 360 days. (3) 60 years are needed to complete 5 revolutions of guru and 2 revolutions of Sani. Thus a cycle of 60 years has been adopted for samvatsara of guru. This is the active life period of a man. (4) Guru varsa are listed in verses 32-46. Varahamihira in Vrhatsamhita has assumed the beginning of samvatsararakra from 35th samvatsara Prabhava, instead of the first vijaya. However, the calculation method given here will start guru, samvatsava from the 13th 'vikrama', for start of first rasi. Thus one complete round of 12 rasis in 12 samvatsaras is considered complete at beginning of guru motion. This is only a convention. Same result could have been obtained by calculating rasi lAean Planets & f madhyama guru and count the samvatsara from 13th. (5) Symbolic formula (a) Madhya guru = B bhagana + R rasi + A amsa etc. (b) Total samvatsara = (B+3)xl2+R+2 = S (c) S/60 - s+r (remainder to 59) (d) r+1 is 1 to 60 samvatsara counted from prabhava. (6) Samvatsara for 1st rasi-completed R=0 / B=0 n = (r+1) counted from 35th samvatsara x « S (B + 3) x 12 + R + 3 12 B + R + 38 + 35 38 + 35 - = = 13 remainder 60 60 Verse 26 - Elapsed part of guru varsa - (Omit bhagana and rasi from madhyama guru). Multiply amsa by 12 and add its 1/330 part which indicates elapsed days of samvatsara. (gata dina). (Deduct it from 361.02672 to find remaining days i.e. bhogya dina) Explanation - 30° of rasi - 361.027 days 361.027 _ 1.027 r = -^-= 12 + -35- days 1° 12 x 1 + 1.027^ _ T 1 ^ = 12 x 11+ 360 330 approx. Verse 27 - If in a Candra varsa, madhyama guru does not move to different rasi, it is called adhivatsara. (Guru varsa is 7 days bigger then Candra varsa and it may not complete 1 raSi in that period. 68 Siddhanta Darpana Verse 28 : If with sphuta gati guru crosses two rasis in a saura varta, then it is called lupta varsa (samvatsara) (Normally guru will touch 2 rasis every saura varsa which is only 4 days bigger) unless both years start almost at sometime within 4 days gap. However, if its true motion is faster, and years start almost same time, it may touch the third rasi also at end of saura varrha) Verse 29 - If in a saura varsa, guru in its sphuta motion goes to next rasi at higher speed (aticara), and does not return to the same rasi, that year is called mahacara kala. This year is as bad and inauspicious as a lupta samvatsara. (In this year also sphuta motion is faster than madhyama gati, not compensated by reverse motion. But guru may not cross into 3rd rasi, if its samvatsara does not start with saura varsa). Verses 30-31 : 60 Barhaspatya varsa contain 12 Barhaspatya yuga (of 5 years each). Divide current number of barhaspatya years by 5, add 1 to the result. Sum is guru yuga starting from Acyuta etc. Within the yuga, the years are named according to remainder as 'sam', pari, ida, 'anu' and 'id' vatsaras. Their adhipatis are agni, surya, candra, brahma and Siva respectively. Comments : This classification of vatsaras was" done in vedanga jyotirha. In one yuga of 19 years, there were five types of years. The years starting from 1st to 6th lunar tithi was called samvatsara. Years starting (solar) from next block of 6 candra tithis were called pari, ida, anu and id vatsaras respectively. In a yuga of 19 years, there were 5 years of samvatsara type. Subsequently in yajur Mean Planets 69 jyotisa, a yuga was of 5 years, each of the 5 vatsaras occuring once. Same names have been adopted for barhaspatya yugas also. Verses 32-46 : Names of barhaspatya yugas, varsa and good or bad years - Yuga (adhipatis) years Subha(s) or Asubha (A) 1. Visnu (Visnu) 1. Prabhava 2. Vibhava 3. Sukla all Subha 4. Pramada t 5. Prajapati 2. Barhaspatya 6. Angira S (Brhaspati) 7. Srimukha S (First yuga 8. Bhanu A according 9. Yuva S to our method 10. Dhata A of calculation) 3. Sakra 11. Isvara S (Sakra) 12. Bahudhanya S 13. Pramada A 14. Vikrama A i 15. Vrsa * * S (Guru will cross vrsa rasi, when vrsa samvatsara will start). 4. Pavakiya 16. Citrabhanu . (vahni) 17. Subhanu 18. Tarana all asubha 19. Parthiva 20. Vyaya 5. Tvastra * * 21. Sarvajit S (Tvasta) 22. Sarvadhari S 70 Siddhanta Darpana 23. Virodhl A 24. Vikrti A 25. Khara A 6. Ahirbudhnya 26. Nandana S (Ahirbudhnya) 27. Vijaya S 28. Jaya S 29. Manmatha A 30. Durmukha A 7. Paitrka • 31. HemalambI • S (Pitara) 32. VilambI S 33. Vikari A 34. Sarvari A 35. Plava A 8. Vaisva 36. Sokakrta • S (Visvedeva) 37. Subhakrta * S 38. Krodhi A 39. Visvavasu A 40. Paravasu A 9. Candra 41. Plavanga A (Nisapati) 42. Kilaka A 43. Saumya S 44. Sadharana * S 45. Virodha krta * A 10 . Aindranala 46. Paridhavi S (Indra and 47. Pramathi S Agni) 48. Ananda S 49. Raksasa A 50. Anala A 11 . Asvina 51. Kapila A (Asvini 52. Kaia A kumara) 53. Siddhartha S 54. Raudra A 55. Durmati A 12 . Bhagya 56. Dundubhi A (Bhaga) 57. Rudhirodgari A 58. Raktaksa A Mean Planets n 59. Krodhana A 60. K£aya A Verse 47 - Surya and Candra complete their full bhaganas in a mahayuga or in a padayuga. Hence their madhyamana can be calculated even from ahargana for mahayuga or for any padayuga also. Verse 48 - Another short method of finding ahargana is described below. It is not a fault for being a repetition, as great poets like Sri Harsa also have adopted such practice. Verse 49 : Multiply years since beginning of creation by 12 and add completed months from caitra sukla pratipada. Keep it in two places. At one place multiply it by 1,00,00,000 and divide by 32,53,55,104. Add the quotient to result in second place. Multiply the result by 30 and add complete days passed after amavasya. Keep it again at two places. At one place multiply it by 1,00,00,00,000 and divide by 63,90,97,35,058. Deduct quotient from quantity in second place. Result will be ahargana from beginning of creation counted from midnight of Lanka. Derivation of Formula Saura varsa x 12 = saura masa Completed Candra masa from caitra pratipada is assumed equal to saura masa. This approximation does not affect the result as the remainders found in calculation of adhimasa or ksayatithi are not used. Total saura masa x 30 = saura dina. Candra tithi after amavasya are # similarly assumed equal to saura dina. 72 Siddhanta Darpana No of adhimasa - T r # - Adhimasa in a kalpa = No. of sauramasa (s) x Saura masa in a kalpa 1,59,33,36,000 _ 1,00,00,000 51,84,00,00,000 X 32,53,55,104 a PP rox ' This is added to sauramasa to get candra masa. candra masa x 30 = candra tithi Ksaya tithis till ista day __ No. of sauradina Ksaya tithi in kalpa till ista day (D) Sauradina in kalpa _ 25,08,22,52,000 _ 1,00,00,00,000 15,55,20,00,00,000 " D x 63,90,97,35,058 ap " prox. We keep the significant digits same, so the approximation is sufficient for knowing integral numbers of adhimasa or ksaya tithi. Verse 50 : For calculating aharganas from kali beginning, the same procedure will be followed. However, 4 zeros from the multipliers will be removed and 4 last digits of divisions (5104 and 5058) also will be taken out. Kaliyuga started on sukravara; so days will be counted from friday. Note : Kaliyuga = 1/10 yuga 1/10,000 kalpa. Hence 4 less no. of digit are required for approximation. Thus multipliers and divisors each are divided by 10,000. Verse 51 - Kalpa bhagana is multiplied by 1811 and divided by 4000 to get bhagana at the end of dvapara. If the madhyama graha calculated from kaliyuga first day to ista day is added, madhya graha from beginning of kalpa is obtained. Derivation : Total yugas in a kalpa = 1,000 Mean Planets 73 Total yugas upto dvapara end 6 manus x 71 = 426 yuga 7x4 14 7 sandhya x satyayuga = = — yuga 4+3+2 9 Satya + Treta + dvapara = — = — yuga 79 Time in creation = — yuga (to be deducted) Hence total yuga upto dvapara end is 14 9 79 1 1811 426 + 27 + _ + ___ = 453 __ = _ Bhagana at dvapara end _ Yuga at dvapara end 1811 Kalpa bhagana Yuga in a a kalpa 4000 Verses 52-55 : Position of graha, at kali beginning (midnight of 17/18 February 3102 B.C. at Lanka) are given below in vilipta (seconds). 12,41,568 Candra mandocca 4,34,160 mangala mandocca 4,56,840 Budha mandocca 8,13,240 gum mandocca 6,01,020 Sukra mandocca 2,35,548 Sani mandocca 8,97,480 guru pata 2,55,960 Sukra pata 1,96,020 Sani pata 3,25,620 At the time of writing Siddhanta Darpana, kali year 4970 end has been taken as reference year (karanabda). Deduct this number from the number of years passed since kali. Add 12 zeros to the right and divide by 2,73,77,85,151. The result will be gata dina from somavara. Ahargana will be from end day of sphuta mesa sankranti (year 1869 A.D.). Mangala 12,41,561 Budha sighra 1,13,724 guru 82,620 Sukra sighra 1,49,040 Sani 11,91,02 Surya mandocca 2,83,176 Candra pata 7,14,788 Mangala pata 1,04,328 Budha pata 1,06,271 74 Siddhanta Darpana Deduction : This is calculation of savana dina in a solar year. In a kalpa of 4,32,00,00,000 solar years, no. of savana dina is 15,77,91,78,28,000. So, savana dina in ista year (D) 15,77,91,78,28,000 " , = 4,32,00,00,000 X ™ of y* ars M or D = y x 15,77,91,78,28 10 12 4,32,00,00 2,73,77,85,151 First day of karanabda was monday. This will be ahargana till completion of year on mesa sankranti of madhyama surya. Verse 56 - Normally madhyama surya enters mesa, 3 days after entry of sphuta surya. So this third day after sphuta mesa sankramana, 1 ahargana or main day of pancanga is taken. Therefore, madhyama graha is to be calculated for previous day of madhya mesa sankranti or on 2nd day of entry of sphuta surya in mesa. Then difference of grahagati for 1 day is to be added for madhyama graha of ista dina. -Verse 57 - There are different practices in different countries. Some pancangas take the entry of sphuta surya in mesa. Many pancangas take caitra sukla pratipada as 1st day. After madhyama saura varsa end, karanabda (4970 kali or 12-4- 1869 A.D.) started. Author has given madhyamanas of dhruva (rasi at the beginning of year), ucca, pata etc. That day was soma vara (monday) and spasta surya had just entered mesa at sunrise. Verse 58 - Now madhyama dhruva (mean constants) for graha, mandocca, slghrocca, pata etc. are stated for somavara day before karanabda at Mean Planets 75 time of sunrise at lanka (0° meridian through ujjain at equator) Verse 59-69 - Table of Karanabda dhruva - (in rasi / amsa / kala / vikala / para) (For 12-4-1869, Lanka sun rise) Ravi Candra Mangala Budha Sighrocca Guru Sukra Sighrocca Sani Ravi mandocca Candra mandocca Mangala mandocca Budha mandocca Guru mandocca Sukra mandocca Sani mandocca 11/28/15/20/46 0/3/20/29/53 5/1/24/17/25 10/18/14/9/2 0/3/45/1/21 11/13/41/42/12 7/18/12/17/24 2/18/47/54/0 10/22/34/59/4 4/7/1/42/13 7/16/4/10/16 5/17/17/0/15 2/5/39/38/29 8/9/19/44/10 Pata dhruva of candra are corrected for reverse movement (bhacakra Suddhi is substraction from 12 rasis) Candra pata (Rami) Mangala pata Budha pata Guru pata Sukra pata Sani pata Ketu pata 3/21/19/18/28 0/28/51/23/4 0/29/17/28/58 2/11/3/15/59 1/24/3/31/0 3/0/13/27/24 9/21/19/18/28 Verse 70 : The dhruva above have been calculated according to proportion ^of kalpa 76 Siddhdnta Darpana bhagana. Candra pata is called Rahu, 6 rasi or 180° away from that is ketu pata. Verse 71 : Method to calculate mandocca and pata for past days has already been described. Mandocca and pata for a particular year can be calculated by this method. Multiply ista varsa by kalpa bhagana and divide by 2,00,000 which will tell the position in lipta etc. _ . t . Ista varsa Ista bhagana Derivation - —7 = — «-* — Kalpa varsa Kalpa bhagana or Ista bhagana - Ista varsa x — -^ ^- 1 - ' * • ' Kalpa varsa Kalpa bhagana x 360 X 60 lipta = Ista varsa x - 2 — : *- — 4,32,00,00,000 Kalpa bhagana = 1?{a var?a x 2,00,000 ' u P a Verse 72 : Add this result to karanabda dhruva (deduct from pata) to get ista graha, ucca, pata etc. Alternatively, this can be calculated from annual motion (hara) also. Verse 73-74 - Hara (annual motion) in lipta is obtained by dividing kalpa bhagara by 2,00,000. Multiply elapsed years after karanabda (gata varsa) and add to dhruva to get ucca, graha etc. Verse 75 - Table of pata hara - Ravi mandocca hara 599 Guru mandocca hara 248 mangala mandocca hara 645 Sukra mandocca hara 359 Budha mandocca hara 488 Sard mandocca hara 2857 Mangala pata hara 671 Budha pata hara 362 Guru pata hara 1818 Sukra pata hara 212 Sani pata hara 367 Verse 76 - (Normally all astronomers assume that mandocca and Sighrocca move from west to Mean Planets 77 east). Author says mandocca of mangala, budha and sani and slghrocca of Budha moves in both directions. This will be discussed while calculating true motion (graha sphuta) Verses 77-78 : While praying to lord Jagannatha in end, author states position of nilacala (Purl temple). It is 284 yojana north of equator on sea coast and 184 yojana east from Indian 0° longitude (Ujjain). Chapter - 4 CALCULATION AT DIFFERENT PLACES Scope - In chapter 3, madhya graha etc were calculated for Lanka. In this chapter, calculations will be done for any place on earth. Mathematical Notes and definitions - r (1) Trigonometrical ratios- L ACB = 0, /.ABC is a right angle Then the , following ratios depend only on the value of angle 0, and not on the lengths of the sides of triangle. By definition these ratios are - Fig. 1 SinG = Cos0 = Tan0 = AB AC BC AC AB BC Cote = Sece = tane 1 cose Cosec e = 1 sin e Fig. 2 Calculation at Different Places 79 (2) Indian Terms - To avoid decimals, a circle of circumferance 21,600 units, i.e. radius of 3438 units is taken. One unit of circumference is equal to 1 kala, then 21,600 kala = 360° = 1 revolution. We draw OA and OB, two radii such that Z.AOB = 6 Jya of LB is AC = R sin or sin x 3438 kala AC is half of the chord AD which is like string of bow shaped arc ABD. Hence its name is Jyarddha or Jya in short. OC is kotijya =R Cos = 3438 x Cos Tangent on A, meets base OB at E. , AE/OA = tan or AE = OA. tan = R tan Hence this ratio is called tangent or tan in short. In sanskrta it is called sparsa jya. OE pierces like arrow, hence called chedjya. OE = OA sec = R sec (sec is short of secant), Complement of angle 6 i.e. 90°- is called koti of the angle. Thus koti jya = jya of koti, koti sparsa jya = sparsa of koti and Koti chedajya = chedajya of koti In sanskrt another ratio is defined, called utkrama jya which is CB = R (1-cos 0). (3) Ratio of circumference to diameter is fixed and is called it (a greek letter, pronounced as 'pai') in modern mathematics. It is a transcendental number which cannot be expressed by any exact number. It can be expressed as non-recurring non-terminating decimal number to any desired 80 Siddhanta Darpana approximation. Values upto 1,00,000 decimal places have been published. Calculation was on computer by the formula -i 1 -1 1 -1 1 7i = 24 tan - + 8 tan 1 — + 4 tan A — 8 57 238 tan" 1 A is an angle such that 6 = tan A. It can be expressed as an infinite convergant series when A is smaller than 1. 22/7 and 355/113 are rough practical ap- proximations of it correct upto 2 and 6 places of decimal respectively. If paridhi is expressed in kala, radius is 3437 3/4 kala approximately, which is same as 1 radian angle. (1 radian is an angle made by arc equal to radius) Madhava of Sangamagrama (kerala) in 13th century used infinite series to calculate value of n up to 30 places and sine table upto 9 places. Value of n up to 30 places have been expressed in a verse by him (read with katapayadi notation) - Accordingly, circumference _ diameter 3.14,15,92,65,35,89,75,43,23,84,52,64,33,83,279 — (4) Yojana - Yojana is a measure of length as explained in the first chapter. Siddhanta darpana takes yojana of 1600 hasta = 24,000 feet or 7.3152 kms approx. (if 1 hasta is taken as 18"). It takes diameter of earth as 1600 yojana then it is about 4.94 miles approximately (hand will be about 19.6"). Calculation at Different Places SI (5) Longtrude, Latitude and sphuta paridhi - Study of sides and angles on a sphere is subject of spherical Trigonometry. It is called gola pada in jyotisa. To know position of a point in space by measuring its angle or distance from fixed point and lines is the basis of coordinate geometry (or cartesian geometry in the name of Rene de-Cartes of France, the originator). In a plane, two systems are used to indicate location of a point. P y ..p y ;y X' X Y X Cartesian Co-ordinates Polar Co-ordinates Fig. 3a Fig. 3b In both systems, O is the fixed point called origin and a line through it OX is called X axis. In cartesian coordinates, another line OY perpen- dicular to OX (in counter clock wise direction) is called Y axis. In cartesian coordinate location of a point P is indicated by its distance x from 6 along axis (x coordinate) and distance y in direction of y axis (y coordinates). Distance in the direction OX' and OY' are negative. (Figure 3 a). In polar coordinates, location of a point P is indicated by its distance r (always positive) from origin O and the angle 6 made by OP with OX in counter clockwise direction. (r,0) indicate posi- tion of any point in space (Figure 3b) Conversion from one system to ofher is not difficult. 82 Siddhanta Darpana -r - jc + y 2 x - r cos 6 = tan -1 y/x y = r sin For example, if Bhubaneswar be origin, then location of Puri can be indicated in cartesian coordinates as 40 kms south (x coordinate) 35 kms east (y coordinate) In polar coordinates - 53 kms away (r) in direction of 40° (0) from south towards east. In a plane, two quantities called coordinates are needed to locate a point. In space, 3 quantities are needed - so it is called 3 dimensional space. In theory of relativity, time is considered fourth dimension. An event in world is indicated by 3 space and 1 time coordinates. Hence world is called 4 - dimensional space time continum. For example, a hill top in Puri can be specified by its height from mean sea level, in addition to two coordinates of plane. Three dimensional coordinates : Cartesian Space Co-ordinates Spherical polar Coordinates F| fl- *a Fig. 4b Cartesian space coordinates are measured along mutually perpendicular X,Y,Z axis. If a right hand screw is rotated from X direction to 'Y direction, it will move in Z direction. The distances of any point P from origin O along the three axis are called (x,y,z) coordinates. Calculation at Different Places 83 In spherical polar coordinates, distance OP of P from origin is r coordinate. Angle 6 between plane of z axis and OP with X axis is second coordinate. In the plane, elevation of OP from XY plane (with line OQ) is called </>. 6 takes values from to 2 n or 360 \ <p takes values from - 90° to + 90° or can take any value. This system is more useful for spherical geometry and astronomy. Conversion formula - r sin 4> = Z, r cos <b cos 6 = x, r cos O sin $ = y In astronomy, only two angle coordinates are used. For places on earth, the distance from centre is fixed as radius of earth (r coordinate). OZ is line from centre to north pole. Angle 6 is measured from prime meridian (great circle or plane passing through north pole and Greenwich (London)). In India, prime maridian was assumed through Ujjain as a reference. O is the angle with equator plane (XY plane). In popular terms 6 Coordinate is called longitude (- 180° to + 180° and <I> coordinates is called latitude (- 90)° (south) to +■ 90 Q (north). Positive direction of longitude is called east, and negative direction west). In astronomy, a second frame of reference is also used. This is fixed with refrence to stars which don't move. Planet's movement is observed with reference to stars. Zodiac or ra& vrtta is path of apparent motion of stars in which coordinates is measured from 0' to 360°. Deviation from this plane is called viksapa or Sara. (-90* to +90*). 84 Siddhanta Darpana For calculation of eclipse etc, frequently we need to convert the figures from equatorial coor- dinates to zodiac coordintes. This is called drk Karma. Sphuta paridhi of earth, at any point is circumference of circle on earth's surface parallel to equator (latitude) circle or simply called a parallel of particular degree. (6) Motion of a top and earth's motion Spin arround axis of the top 5a Spin with steady precession of axis in a vertical cone 5b Fig .5 Spin, precession and nutation 5c A top rotating fast along its axis stands vertical on a rough surface due to gyroscopic stability. Its lower end is fixed due to friction with earth and it moves away from vertical position and falls due to gravity in the end. Spin (figure 5a} - Rotation of a top about its axis is called spin. When top is rotating very fast, its axis is vertical and its appears stationary. Precession (fig 5b) - Precession is conical motion of the axis of top. Upper point of the axis makes a circle about the vertical direction. Nutation - When motion of top becomes slower, its axis falls further away from vertical and rises again alternatively. In steady precession, upper point of the top makes a horizontal circle on a sphere. In nutation it moves in a wave like Calculation at Different Places 85 th between two horizontal circles on the sphere as in fig. 5c. (7) Rotation of earth around its axis - Motion of earth around its axis is completed in one day and causes day and night. Due to that the sphere of stars in sky appears to make a daily rotation from east to west. This is spin motion of a top. Axis of earth is inclined at angle of about 23-1/2° from perpendicular to the plane of ecliptic (i.e. plane of earth's orbit round the sun). Due to that the sun appears either north or south of the equator. During summer season in north hemi- sphere, it will be perpandicular to earth's surface at noon time at some place between equator and 23-1/2° north (Tropic of cancer) When the plane containing vertical to ecliptic and earth's axis contains sun, inclination of sun towards north or south is maximum. These points opposite to each other are called summer and winter solstice. In summer solstice, axis is directly inclined towards sun, and sun is perpendicular to tropic of cancer (23 1/2°) Autumnal equinox Summer solstice N Winter solstice Vernal equinox Fig. 6 At two points on orbit, 90° away from place of maximum inclination, the axis of earth is inclined side ways and not towards sun. Then sun rays *** perpendicular on equator (i.e. on plane con- 86 Siddhanta Darpam taining ecliptic and arth's axis). On such points, day and night are equal. 'Nakta' means night in Sanskrit, it is called noct in greek. Equinox means equal day and night. On one of equinox points, sun goes from south to north hemisphece. This is called vernal equinox. The other point is called autumnal equinox. Northward motion of sun is called uttara - ayana and southward motion is daksinayana. Both ayanas, make one hayana, a complete year. Precession of axis - At present, earth's axis towards north is directed to pole star (Dhruva Tara). So pole star appears to be fixed. Axis is moving like precession of a top in conical motion due to two reasons - (1) Earth is not spherical, it has bulge at equator due to centrifugal force of rotation (2) Orbit of moon is inclined to earth's orbit at about 5° angle which creates unequal pull at different ends of bulge. To some extent, inclination of other planetary orbits also affects the axis. Practical effect of precession of axis is that, points of equinoxes move slowly westwards. If solar year is counted by motion relative to fixed stars, start of seasons shifts slowly. V change of equinox, i.e. 1 day change of season occurs in about 72 years. One month change is in about two thousand years. In western astronomy, solar year is counted from equinox to equinox. Position of vernal equinox is taken as 0° mesa. Difference between vernal equinox, and static mesa 0° of Indian astronomy is called Ayanamsa. For determining day length, rising period of rasis etc, position of sun from Qdadati * 1 87 eqUhl >ll ? triangle is completed. Since equinox SP ^ backward (to west), ayanamsa is added to ^position. It is caUed sayana sun or any other planet. Y REFERENCES 1. For trigonometry, any school text book can be referred like S.L. Lone/s Trigonometry. 2. Cartesian geometry of two dimensiouns can be found in any college text book, e.g by Loney or by Santi Narayana. Geometry of 3 dimensions can be found in book by R.J.T. Bell. 3. Results of spherical trigenometry can be found in text books by Todhunter or by Gorakh Prasad. 4. Transformation of axis can be found in books of classical mechanics or foundations of vector/tensor analysis. Differential geometry of Weatherburn or by Shanti Narayan can be referred for space curves, surfaces and polar coordinates. 5. Polar coordinates/transformation of axis are explained in classical mechanics also. M.Sc/Hons level text books also discuss mo- tion of top. The following books may be referred. Classical Mechanics - by Goldstein. 88 Sictdhanta Darpan Principles of Mechanics - by Synge & Griffith Mechanics - by Simon Earth's top tike motion has been discussed i detail in motion of top (4 vols) by W. Sommerfield & Felix Klein Translation of the text (Chapter 4) Verse 1 - I (author) will describe in short the various measurements of earth. In second half of the book, these will be discussed in detail. Verse 2 - Average diameter of earth (madhyavyasa) is 1600 yojanas. Multiply this byj 10,800 and divide by 3,438. You get the paridhi (circumference) described in 3rd verse. Verse 3-4 - Paridhi at centre (equator) is 5,026/10 yojana. Jya of 90° is taken as 3438 kala. Hence, sphuta bhu-paridhi is obtained by multi- plying, madhya paridhi by lamba jya of the place and dividing by 3438. Otherwise, this madhya paridhi can be multiplied by 12 and divided by visuva karna. RQ. 7 Derivation - (1) NS is line joining north and south pole. O is centre. The circle perpendicular to NS line is called sphuta bhu paridhi. Largest circle passes through centre 0, at point A and is called equator. Sphuta paridhi at point P is to be calculated. addition at Different Pious 89 OA = OP = radius R of earth Paridhi at centre is 2 tz R = C Latitude of place P is Z.POA = (Aksamsa) Lamba amsa = 90° - 6 = ZPOD = <D For circle of sphuta paridhi at P, r = DP = OP sin 3> or r = R sin O Circumference = 2tfr = 2jrRsin<l> Lamba jya 3438 =C sin <P = C RsinO = C x R (2) Second method is based on measurement of palabha explained in Triprasnadhikara. On Visuva samkranti, sun rays are perpen- dicular on equator, i.e. paralled to OA. At point P, a pole PR is kept vretical of 12 unit lengths. Its shadow PC on horizontal surface is palabha and RC is Pala Karna or visuva karna. In Fig 8, OPR is straight line, RC I I OA or RPC and ODP are similar. Fig. 8 Hence RC OP or r = PR PD 12R or Visuva Karna 12 • * ^ R ■ ~ r hence (he result. Visuva Karna • ■ Verse 5 - Lanka, Rohitaka, Avanti, Kuruksetra etc. are on the prime meridian line (Pradhana madhyandina rekha) which passes through both merus. * 90 Siddhdnta Darpawt Note (1) Rekha is a straight line in a plane but it is arc of a great circle in a sphere (the circle passing through centre of sphere, which is greatest). Like straight line of a plane, it is the shortest distance between two points, and doesn't! change the direction. ,| (2) . This verse means same as verse 62 in \ madhyamadhikara of Suryarsiddhanta and repre- sents the convention of treating the longitude through Ujjain as reference line (0° longitude). At present, the meridian passing through Greenwich is 0° meridian. (3) According to historical traditions, 'Polaris narua' (meaning Paulastya nagara) in present Sri; Lanka was the capital of Lanka. However, for astronomical purpose, Lanka is the imaginary point of intersection of longitude through Ujjain and equator (i.e. middle point of that line between south and north pole). Lanka is nearest land mass near the point; hence it is called Lanka (presump- tion) (4) Location of original Kuruksetra is not known. If present Rohataka (a district headquarter in Hariyara) is taken as Rohitaka, then it is 8 pala east from madhya rekha. Hence, Bhaskaracarya has not indicated it on madhya rekha. He says that this line touches regions like Kuruksetra etc. Verses 6-9 - Desantara is the east west distance between two places with same aksamsa on sphuta bhu paridhi (local latitude circle perpendicular on polar axis or parallel to equator). Multiply this desantara yojana by 60 and divide by spasta bhuparidhi. Alternately, multiply Calculation at Different Places 91 by visuva karna in liptikas and divide by 60, 314. You will get desantara in danda etc. All days, months and years start with mid- night at Lanka i.e. from midnight at places on madhyandina rekha. If a place is east from rekha, add the desantara (ghatl ) to get the midnight time at that place, from which day, months will start at that place. If the place is west from rekha, desantara is to be deducted. Derivation - (1) Earth rotates with uniform speed around its axis or in the direction of bhuparidhi. Complete rotation of bhuparidhi takes 60 danda or 1 day. Thus by ratio and proportion Desantara in danda 60 danda Desantara in yojana spasta bhuparidhi or Desantara danda = Desantara yojana (east west distance) sphuta bhuparidhi (2) Visuva Karna - I*alabha is length of the shadow of a vertical stick (cone or Sanku) at noon on a day when day and night are equal. Height of sanku is 12 angula. Visuva karna or pala-karna is the length of hypotenus, i.e. distance from tip of 12 angula sanku to the tip of shadow. Palabha or pala karna gives a measure of the angle of latitude (aksamsa) as sun is vertically above equator on visuva day (when day and night are equal) In Figure 9, X is a place on aksamsa 6\ Angle of sun rays at mid day will be 0° at equator, ^o 92 Siddhdnta Darpana 12 anguia Fig. 9 it will be 6° at latitude 0* (Derivation 2 after verse 4, Fig 8) i.e. AXYZ = 6 (aksamsa) XY is Sanku of 12 anguia (units) of length. XZ is palabha and YZ is palakarna. Sphuta bhuparidhi = 2 jt r (r = sphuta Trijya) = 2 jt R cos 9 (R = radius of earth) XY = Bhuparidhi x ^r= __ Bhuparidhi x 12 anguia ~ Palakarna anguia desant ara yojana x 60 DeSantara dan^a = sphuta bhOpandh. desantara yojana x 60 = *-f — — x palakarna Bhuparidhi x 12 v desantara yojana x palakarna in lipta 60314 (As per verse 2, bhuparidhi x 12 = 5026/10 yojana x 12 = 60314 yojanas) Verses 10-11 - Some astronomers opine that day starts everywhere from the sunrise at Lanka. Due to that confusion, the author decides that at any place the lord of vara will be ruling from sunrise at that place for period of 60 dandas. Verse 12 : Bala (power) of yama and yamardha is not connected to siddhanta (astronomy) it is -A Calculation at Different Places 93 useful for phalita (astrology only). So it is not discussed here. Verse 13-14 - Bhaskaracarya (and his followers) assumes start of all (motion of planets, day etc.) from sunrise at Lanka. Thus the ahargana according to his theory is different from other theories. This separate ahargana (of Bhaskara) doesn't give position of planets as they are actually seen, hence it is not followed in this book. Note - Bhaskara ahargana will give correct position of planets for sunrise at Lanka only. Since day length is different for different latitudes, sunrise will be at different times on same longitude also. But midnight will be at same time on the whole longitude, hence it gives correct result. Verse 15 - Method of finding midnight position of planets at ista (desired place) - Multiply desantara kala (in danda) of the place with dainika gati of graha and divide by 60. Add the result to the graha at Lanka at midnight if the place is west from Lanka. (Since earth rotates in east direction, midnight will be later in a place to the west and in the extra time, the graha will move further). Deduct, if the place is towards east. Verse 16 - Alternately, difference in grahagati can be obtained by multiplying dainika gati with desantara yojana and dividing by sphuta bhuparidhi. Note : Desantara ghati of a western place is the time taken by earth to reach midnight position for that place. Alternate method follows from methods of hndig desantara ghati (vrse*9). i i 94 Siddhanta Darpam Veise 17-21 - Old method of finding longitude - calculate the time of purna (full) candra grahana (lunar eclipse) at madhya rekha (prime meridian through Lanka or Ujjain). (Note - Exact time of Purna grahana is the time of unmilana (when moon starts emerging from shadow). By observation, see the actual time of Purna grahana at your place. The difference in time is desantara kala. If the place is west from Ujjain, then the time found by observation (drk-siddha or vedha) wul be less than calculated time (i.e. eclipse will be at same time, but corresponding local time will come later at western place). For places east of Ujjain, observed time will be more. Time difference can also be calculateed on basis of sparsa (when moon starts entering the shadow) or moksa (when moon completely emerges from shadow). To find desantara yojana, multiply it (desantara kala) by sphuta paridhi and divide byj 60 (already explained in verse 9). j To calculate graha at ista time, multiply the] dainika gati of graha by ista kala and divide by 60. Add the result to graha at midnight at the place. Notes : (1) Time difference (in danda) frorr^ Lanka midnight is due to two components - (l)i difference between midnight times at the place ancl at Lanka (2) Time lapsed after midnight of the place at desired time. 95 Calculation at Different Places Dainika gati of graha is movement in 60 danda (1 day). Hence movement in ista danda is Dainika gati x ista danda (kala) 60 components of ista kala are added or sub- stracted as explained before. (2) Candra grahana is due to covering of moon by shadow of earth, both of which are at one place. Thus there is no parallax and it is seen similar from all positions. But Surya grahana is by obstruction of sun's vision by moon (at i/400 of the distance). Their relative directions are seen different from different places, (called parallax), hence surya grahana starts at different places at different times. Hence only candra grahana can be used for comparison of midnight times. (3) Terms of grahana Fig. 10 In fig 10, S is sun, E is earth and M is orbit of moon. C is shadow cone of earth due to rays from sun. 1,2,3,4 are successive positions of moon. 1. position of moon touching the shadow - at sparsa kala. 2. position of moon when it has just entered completely in shadow - Nimilana or sammilana (meaning closing of eyes) kala • % Siddhanta Dnrpana 3. Position of moon about to emerge from shadow; unrmlana (opening of eyes) kala 4. position of moon when it has just emerged completely from shadow - Moksa kala Grahana will be discussed more completely in chapters on candra and surya grahana (4) Other methods of finding longitude - Now very accurate watches are available and any event can be observed with telescope more accurately. In observing candra grahana, there will be difference of 2-3 minutes in observation by different persons. Eclipse of satellites of jupiter occurs daily. It is observed through telescope and compared with time given in nautical almanc. This will give accurate longitude. Alternatively, two watches are to be tallied with local times of places, whose longitudes are to be compared. They can be tallied with sunrise or preferably at midnight time. Then by telephone, the local time of the two places can be compared. The time difference will be desantara kala. Nowl T.V. and radio announce Indian standard times (mean time at 82° 30' east of greenwich). Local mean time can be found by correcting local true time! with time equation (fixed for particular days ofj solar year or sun position). From that time] difference, difference with 82° 30' longitude can be known. (5) Time can also be known accurately b movement of stars during night. This is particular]: useful for sea journeys in a clear night. Since, method .of finding longitude was known since remote past in India, long journey in sea wa Calculation at Different Places 97 possible. Due to difficulty in knowing time in absence of watches, this method could be known in western astronomy only in 1480 A.D. after which Cobumbus could undertake his journey, in 1492 in pursuit of sea route to India from Spain. Finding latitude is easy through palabha, discussed in more detail in Triprasnadhikara. Verses 22-24 : By above corrections for desantara kala, we get the graha for niraksodaya kala (sunrise time at equator at same longitude). Due to difference in aksamsa (north south distance) from Lanka, cara samskara is needed, because sunrise times are different for different places on same longitude due to aksamsa. From sphuta ravi (sun) kranti (true inclination of sun from vertical in north south direction i.e. inclination from vertical at noon), find cara danda (time in danda by which day-half is longer than normal day half of 15 danda). Multiply it with dainika gati of graha and divide by 60. If sun (sayana) is in six rasi from tula to mina, add the result to the position of graha. If sayana sun is in mesa to kanya, then deduct the result. For finding graha at the time of sun set, do the reverse process. Notes (1) This part (chapter 1 to 4) is madhyamadhikara, dealing with mean position of planets. Nothing has been so far discussed, as to how, true (sphuta) position of planets can be found. Sphuta kranti of sun can be found only at moon time by direct observation. By comparison with previous days kranti, it can be calculated for sunrise time (3/4 of the difference of 1 day kranti will be added to previous noon figure? to find kranti at sunrise). 98 Siddhdnta Darparta 2. Mesa to kanya - 1st six rasis are in north hemisphere and other six are in south (sayana rasis to be more accurate). When sun is in southern hemisphere, days will be smaller in north hemi- sphere compared to night. Hence sunrise will be later and sunset earlier than equator (where day night are always equal) Thus graha will move for more time at sunrise compared to sunrise at equator, difference of motion will be added. 3. Cara is variation of day from 30 ghatika, caradala is half of cara. In short cara is used for caradala which is directly calculated. Jya of cara (angular difference in earth's rotation) is called cara jya. V A Fig. 11 4. Explanation of cara - (difference in day, length) O is the place for which it is to be found out for how long, a graha will be above horizon.; NOS is North south line (ksitija rekha) POP' is the north south line at equator (Z.PON is equal to aksamsa of O). NPVSP' is yamyottara vrtta, i.e. the vertical circle in the plane of longitudinal circle (great circle passing through north pole and vertical at place O). Calculation at Different Places 99 A planet in kranti vrtta appears to move daily • a vertical circle at equator in east west direction. Its diameter BOB' is perpendicular to north south r e P'O P at equator. This circle is called ahoratra vrtta (only diameter is seen in perpendicular^plane). Corresponding to point O, the planet rises in the east goes upto B, highest point in sky (south from vertical in north hemisphere) and sets in west again at O. Motion from O to B' and back to O are not visible as these are below the horizon. Both motions OBO or OB'O take 12 hours each. CK'C is the diameter of ahoratra vrtta (diurnal circle) of a planet in south hemisphere. At equator, it is visible for motion K'C K' for half the day i.e. 12 hours. However, at place O, it rises only at point M' and is not visible for period K' to NT (in 12 hours) which is called car a. Time for K'C = 30 ghati (12 hours) For K'M' in morning and M'K' in evening, sun (or a planet) will not be visible above horizon. Thus length of the day is 30-2 K'M' A MA' is diameter of ahoratra vrtta of a planet in north hemisphere. Kranti of planet corresponding to AA' is AB (north) and corresponding to CC it is BC (south) Carakala is time corresponding to movement between KM or K'M' (called ksitijya or kujya) Radius of ahoratra vrtta is called dyujya ('Dyu' means light) Carajya of planet is projection of kujya on visuva vrtta BOB'. It is OR' for north kranti and OR for south kranti. 100 Siddkdnta Darpanw Angle made by carajya (length of circum-! ference) at the centre expressed in prana is called! caraprana or carakhanda. 5. Methods of calculating carajya (Chapter 6-104, p-352) 12 Artgula Fig. 12 AB is cone at a place with aksamsa 0°. It ii kept vertical on day of equinox at noon time. Sino sun rays are perpendicular to equator or that day; it will make angle 0° with AB. BC is shadow at that time (figs 12) L BAG =0 Length of AB is 12 angula as per convention BC is palabha. tan0 = BC Palabha ■(1) AB 12 Now according to figure 11 in para(4), BA i north kranti. ZBOA is angle of kranti (angle no shown) AL is kranti jya (AL J_ OB) AL = OK Now ^KOM = = aksamsa KM Ksitijya Tan 6 OK From (1) Tan = Krantijya Palabha (2) :\ 12 Calculation at Different Places 101 Krantijya x Palabha so, Ksitijya = ^ (3) PKO and PMR' (grand circles) are both perpendicular on AK and BO. Due to similarity of s D herical triangles (as in plane triangles) AE = BO KM OR' BO x KM _ Ksitijya x Trijya or carajya OR' = —^ - ^7 (6) The difference in planet motion at sunrise is calculated by proportion of motion in carakhanda compared to dainika gati in 60 danda. Verse 25 - The value of cara danda for a particular sphuta surya previous year will be same for the equal rasi of madhyama surya this year (exactly same for equal sphuta surya). This ap- proximate equality is used for checking the results obtained through palabha. By taking this value of cara danda, there will be negligible error. Verses 26-30 - Bhujaphala samskara - Now, I tell about another samskara (correction) in madhya graha based on niraksa lagnamana and ayanamsa etc. Mid-night calculated from madhya ravi is different from midnight of sphuta ravi. Difference between sphuta and madhyama ravi is called bhujaphala and correction for that is needed. Add ayanamsa to madhyama ravi, find manda bhujaphala, multiply it by udayasu (time of rising of rasi in prana) of the rasi at equator (niraksa) and divide by 1800. Multiply the result by dainika gati of graha and divide by asu of madhya ravi savana dina. The result in kala etc is to be added °r substracted from madhya graha for bhujaphala samskara. (There are 21659 asus in a madhya savana dina). 102 Siddhdnta Darpana For correction in slghraphala, mandocca of candra or bhujantara of rahu, reverse is done, (positive bhujaphala is to be substracted or vice versa) Notes (1) Manda bhujaphala is neither ex- plained nor method of finding it has been described in madhyamadhikara (chapters 1 to 4). Manda bhujaphala is the correction to graha rail due to its unequal speeds which is slowest at mandocca. (Since sphuta graha is closer to man- docca than madhya graha, it is termed as attraction of mandocca). Real motion of earth E is in an ellipse around sun S at one of the focus. The farthest point E on far side of major axis is the slowest point called mandocca. (It is manda = slow and highest = Ucca ) E, is closest to sun called the nica point. Midd e points of the orbit Ex and E 3 are not at right angle to direction of major axis but towards mandocca position (apparent attraction towards it). Apparent elliptical motion of sun around earth Fig. 13b is explained by combination of two circular move- ments. Fig. 13(a) is real orbit of earth round sun Calculation at Different Places 103 Fie 13b indicates apparent positions of sun calcu- lated by combination of two circular motions. E is earth around which madhyama surya M is moving in a circle in anticlockwise direction. 8 positions are indicated as Mi, M 2 — M 8 . Sphuta graha S is rotating in a smaller circle (manda paridhi) in opposite direction. Both complete the rotation in equal time. Corresponding positions of sphuta graha are indicated by Si, S2 S%, At position 2 for example ZS 2 M 2 V 2 = Z-M{E M 2 as speeds of madhya graha and manda graha are equal. Apparent position K 2 on kaksa vrtta is sphuta graha. M 2 K 2 is called manda phala. S 2 V 2 perpendicular on manda trjya is called manda bhuja phala (fig 13 c) which is almost equal to man- daphala as mandaparidhi is very small compared to madhyaparidhi. M 2 V 2 is kotiphala. Fig. 13c Mandaphala and bhujaphala is negative in 1st semicircle after mandocca. (it is to be substracted from madhya graha). In 2nd semicircle it is positive. Kaksa vrtta is 360° or 21,600 kala. Manda paridhi is expressed in angle in proportion to length of kaksa vrtta. Sin L$ 2 M 2 V 2 = Sin LM& M 2 or - V * s Ul ft S *M 2 *" EM 2 104 Siddhanta Darpana S2 M2 (Manda Trijya) Bhujaphala S2 V 2 = M2 P2 x ^ (kak§a Trijya) man daparidhi (sphuta) = Bhuja jyS x ^ " m andaparidhi Kotiphala = Kotijya x ^j Mandaparidhi also changes slightly, because, earth is not at centre of orbit, but on one side at the focus. (2) Udaya kala of different rasis is calculated in chapter 6-121. Due to oblique direction of rasis with equator (24- or 23 '27' more accurately), the time taken by different rasis to rise is different. As we move away from equator this inclination with local horizontal plane increases. Difference in rising time of rasis becomes more. However, total time of rising of all rasis will be same as Naksatra dina for all places. The rising time Of rasis for 1st to 6th rasis is same as that of 7th to 12th rasis m reverse order. At equator, position of 1st to 3rd rasi is same as 6th to 4th rasi (symmetric for sayana rasi), hence their rising times are same. For difference in start of mid night at ^uator, only the rising times at equator are needed. A com- parison of traditional rising times based on Surya siddhanta and modern values is given below- .'< i Sayana rasi Mesa Vrsa Kanya Simha Tula Surya siddhanta Parama Kranti 24* Mina Asu Vrscika Mithunal Karka 1 Dhanu Kumbha Makara 1670 1795 Fala 278 299 Minutes New observations Par ama Kranti 23*27' Asu |Pala|Minutes 111 120 1935 I 323 I 129 1675 1794 1931 279 299 322 111.7 119.6 128.7 1 At other places, udayasu of rasi is lessened by carasu. It is added for rasi 4 to 9. Calculation at Different Places 105 (3) Since Udayasu is calculated for naksatra dina and dainika gati is calculated as per savana dina of 21659 asu. au dainika gah gati in 1 asu = 21659 ud ayasu x dainika gati gati in udayasu = 21 659 Hence correction for manda bhujaphala manda bhujaphala X gati in udayasu = '~ 1800 ~~ *. because udayasu is for rise of 30° i.e. 1800 kala. Verse 31-32 - Alternate method for bhujantara sanskara / . Ravi manda bhujaphala x dainika gati Bhujantara = ^ 21,600 : = mandabhuja phala -^ (21,600 -5- dainika gati) It will be added or substracted as before. Note - In this formula, different rates of rising of rasis and difference between naksatra dina and savana dina are ignored. Verse 33 : After, bhujantara samskara, I am telling the method of udayantara samskara which is due to difference between madhyama ravi in kranti vrtta and imaginary madhya ravi in nadivrtta (in plane of equator). Verses 34-37 - For this purpose (for udayantara samskara) make the madhyama ravi sayana (add ayanamsa). Find the bhukta asu of that rasi (part of udayasu of rasi in proportion to lapsed degrees in that rasi). Add the udayasu of previous rasi starting from mesa. Then calculate the kala of sayana ravi and substract from 1st result. 206 Siddhanta Darpana Multiply the difference by the dainika gati of graha (in lipta) and divide by 21,659 (as dainika gati is for savana dina of 21,659 asu) The result is udayantara phala. Substract the result from madhya graha, if ravi is- in sama or even pada (2nd or 4th quadrant) and add if ravi is in vteama pada (1st or 3rd quadrant - to 90° or 180' to 270° from mandocca). For correction in pata or ucca, do the reverse. Notes : (1) Madhyama ravi + Ayanarhsa = Sayana madhyama ravi = S Bhukta asu for S = rising times for rasis from 0° to S 1 asu time = time for movement of 1 kala at equator Hence Bhukta asu of S = Its kala at equator = E Kala of S = S' Correction for observation in plane of equator = E-S' in kala equivelent to asu time. Difference in madhyama graha = Dainika gati (E-S) x 21659 as 1 day is of 21659 asu (savana dina) This difference is negative for 1st and 3rd quadrant i.e. Sayana ravi is more in kranti vrtta than in nadivrtta. At 90° and 270° they are equal, and no correction is needed. (2) This is effect of transformation of coor- dinate axis from ecliptic to equator, because time is measured by movement along equator (asu is V movement) . Calculation at Different Places 107 Verse 38-39 - The three samskara (cara, hhuiantara and udayantara) can be made to sphuta Taha also instead of applying it to madhya graha. Then we will use sphuta graha instead of madhya ha in a ii places. Once the samskara has been done to sphuta graha, it is not to be applied again to madhya, mandocca and sighrocca because these results are used to calculate sphuta graha. Note : The sanskaras are for difference in time measurements and not due to madhya or sphuta graha, hence correction to any value can be done. In short, correction time difference between mad- hya and sphuta is negligible. Verses 40-41 - If we take asu arising out of mandaphala of ravi while making bhujantara samskara, then udayantara karma is done from sayana madhyama ravi. When we take asu equal to kala of man- daphala of ravi then udayantara will be done from sayana ravi before bhujantara sanskara, both are to be done separately. Note : (1) In taking asu equal to mandaphala, it is already converted to value in equator plane; hence separate udayantara samskara is not neces- sary. (2) A review of all corrections - (a) Desantara sanskara - It is due to different times of sunrise which is earlier in east. Hence time in east is more counted from sunrise or midnight. At present reference is not 0° longitude only. Every country has fixed reference time according to time zone from 0° longitude through Greenwich. Thus Indian standard time is standard time for 82 * 30' east of 108 Siddhanta Darpana greenwich, i.e. 5-1/2 hours more. Correction for local standard time is done for difference in desantara (longitude) Since 360° rotation of earth 24 X 60 . . . . in 24 hours/ 1° rotation is in 36Q = 4 minutes. Hence 4 minutes time is added for each degree longitude towards east. Local standard time - Indian standard time = (longitude - 82° 30') in degrees x 4 minutes (b) Cara sanskara - Midnight or midnoon is same for all places in a longitude. When time is measured from midnight (in hour system), then no correction is needed. However, in India, savana dina is counted from sunrise which is different at different latitudes. Difference in day length in- creases as we move away from equator. In practice we do not correct the time, but find the time of sun rise. Time of sunrise depends on position of true (sphuta) sayana sun which is fixed for a particular day of a solar year like christian era. It also depends on latitude of the place. Thus date wise charts are prepared for sunrise time at different longitudes (in local mean time), at 1° or 10° intervals. It can be calculated from kranti of that day noted from pancanga or calculated from sayana ravi. • Difference between true time and sunrise time - both counted from midnight gives ista kala in Indian system. (c) Bhujantara sanskara - This is due to difference in standard time and true time - both. Local standard time is calculated on the assumption that each day is of 24 hours. Day length is made Calculation at Different Places 109 < two components. To move from 1 naksatra to irnaksatra again it takes 23 hours 56 minutes a to earth's daily motion. Meanwhile, sun also ^rfves about 1° ahead due to orbital movement of Sin S ame direction (360° in about 360 days). To cover that distance more earth takes about 4 minutes more (360° is covered in about 24 hours). Thus naksatra dina is 23 hours 56 minutes = 21,600 asu and savana dina is 24 hours = 21,659 asu. Difference is 59 asu = about 4 minutes (= 60 asu). While naksatra dina is fixed, extra 4 minute component varies and each savana dina is not 24 |purs exact. But the watches are calculating 24 hours for each day according to standard time. The Standard or mean time and true or solar time start together at sayana mesa sankranti, 23 March., when *Jay and night are equal. Around 24th April when sun is at farthest (mandocca is at nirayana mesa §0° or sayana 32°), sun is slowest. So days are |*naller than 24 hours after 23rd March. By taking |4 hours for each day clock time is slower than true time. This addition in clock time to get true time accummulates for about 6 months upto 14 minutes. Then it is negative correction and again both times tally on 23rd March next year. Effect of 4 minutes shorter naksatra dina is that a particular lagna (e.g. mesa) will start 4 minutes earlier on next day. Effect of difference in true time and standard time is that sun will be at top most position at true noon not at lqcal mean noon (12 hrs local standard time). This is also called correction due to time difference, or velantara sanskara. The formula for knowing difference in true and standard time is called time equation. i^'i : 2 m\ 110 Siddhanta Darpana This difference depends only on sun's position (indicating bhujantara) or the day of solar year. (d) Udayantara sanskara - This is negligible and is not necessary when bhujantara is measured in asus. In modern astronomy also, this is included in time equation. Verse 42 - Multiply aksamsa kala by bhuparidhi and divide by 21,600. Then we get the distance of place from niraksa (equator) towards north or south on the yamyottara vrtta (longitude line). Note - Bhuparidhi covers 21,600 kala. Aksamsa kala is north south distance from equator in kalas. Thus the distance from equator is calculated because 1 kala is same on longitude line or equator. Verse 43-45 - To save enormous labour in calculating graha, I am giving 'padaka' of surya etc. like Kocanacarya. ('Kocanna' was an astronomer of Andhra Pradesh who had prepared charts for easy calculation. These charts were popular in south Orissa also at the time of author). ^ Ahargana is given for years 1,2 — — , 10,20, — , 100, 2000 , thousands, lakhs, ten lakhs, crores and ten crores. These start from madhyama surya at mesa sankramana. Vara Suddhi has been done in this. To calculate the ahargana, add the figures .given in table and divide by 7. If correct vara doesn't come, then add or substract 1 for tally with vara. Verses 46-51 - By this method, ahargana for first day of panjika is calculated. Graha is calculated for that ahargana from their respctive padaka (tables). In this addition, we take figures upto 5 -J4- - 5r-. : . OOculation at Different Places 111 divisions from rasi (para). By this, graha gati can £e calculated for up to 1 arbuda (10 8 ) days. After writing padaka of graha and ucca etc, their dhruva (starting position) at beginning of kali and beginning of Karanabda (standard year for start of calculation by author-mesa samkranti of 1869) are written. Also write the dainika gati of graha, ucca and pata. Write bhujantara, cheda (part) mandocca hara (part), patahara and desantara fcala etc. In the 73 tables, while adding rasi etc of aha, multiples of 12 rasi (1 revolution) are jducted. When calculation is from kali beginning, ~§ge get madhyama graha etc for Lanka midnight. |f calculation is from karanabda; then value is for sunrise. For madhyamana of candrapata (rahu), the angles are deducted from dhruva. Result is deducted from complete revolutions. Verses 52-55 - After obtaining madhyamana, fle§antara samskara etc. are done. Then ahasphuta is done with help of table of landaphala. As a siddhanta grantha, the tables should have been given after their related text. But at the time of printing (in 1899) all charts were #ven in appendix. Verse 56 - For convenience in grahasphuta, I (author) have given phala, dhruva, gati etc in chart for 1 to 10 8 days. After calculating graha sphuta according to charts, you may not observe the graha m same position. Then correction is to be made y seeing dainika gati, dhruva padak etc. in second P«t of this book. in u ISe 57 " k* a y tne ^ orc * Jagannajha reside m y heart who is worshipped by Kubera's friend * 22 Siddhanta Darpana Siva at Nflacala situated at aksamsa 4/27 (palabha) and desantara 8434 viliptas which are 4/19 palabha and 9138 vilipta according to new calculations. Verse 58 - Thus, ends this fourth chapter written by Sri Candrasekhara born in renowned royal famfly of Orissa. Siddhanta Darpara is for tally of calculation and observation and education of students. Padaka charts have been given for fast calculation. 1 i t$ ■ 'i B. SPHUTADHIKARA Scope - This part deals with finding true position of planets. So far we have calculated methods of finding mean position, which assumes constant average speeds of planets, to a first approximation. This part contains two chapters. Chapter 5 discusses true friotion of planets. Chapter 6 deals with special corrections to moon's motion and accurate pancanga on that basis. Chapter - 5 TRUE PLANETS (Making grahas sphuta) General Introduction (1) Concepts of Plenetary motion Figure 1 (a) Figure 1 (b) 114 Siddhanta Darpana Copernicus S represents the sun and P the planet The line KL turns with the angular velocity of the planet round the sun, while LP turns at twice the rate. The length KL and KS is specified for each planet. (Not drawn to scale) Ptolemy £ is earth, P is planet Line A L turns with mean angular velocity of the planet round the sun. The line LP Turns with the mean angular velocity of earth. Length AC = CE is specified for each planet Length LP is related to the earth sun distence (for outer planets) From the data collected over centuries, ap- parent circular motion of planet with some loops and retroacting motion were detected, where it was difficult to find a pattern. But Hipparchus (100-120 BC) and Ptolemy (85-165 AD) were able to describe it on the basis of epicyclic moitons. As explained in diagram of Ptolemy (Fig la) planets moved in circles whose centre moved on some other circle round the earth, centre of this circle was slightly different depending on changes in the velocity of planet. This was successful in predicting the future position of planets, but was unable to reveal any law of nature. Copernicus modified the pattern with similar construction; (Fig lb) but with sun at rest in which patterns were easier to detect. Based on this construction, Kepler (1571-1642) framed 3 laws- P Finurft 2 - Kpnlarian Orbit True Planets H5 Line SP joining sun S to a planet P sweeps out equal areas in equal time intervals (rule 2). P moves on an ellipse with S at focus, (rule 1). OA is semi major axis and ratio OS/OA is called eccentricity. Third law is that square of time period T of revolution of a planet is proportional to cube of its mean distance from sun. These laws led Newton to prove that all matters attract each other with a force proportional to inverse square of the distance between them. Together with plausible assumption that force is proportional to masses of attracting matter, it formed his theory of gravitation. However, the method of calculation of planetary position remains the same. In both the methods, we calculate the direction and distance of planet from sun (heliocentric position). Then on basis of earth's distance and direction from sun, we calculate the direction of planet from earth (geocentric position.) Heliocentric position is only a mathematical necessity. Actual observation is always from earth, equal to geocentric position. (2) Calculation of planets in Western astronomy - Calculation of sun's position is simplest. We calculate position of apsis (nearest point on major axis — Indian method starts with farthest point) mean anomaly (angle with apsis) a nd position of vernal eqinox from which lon- gitudes are measured. In a solar calender, sun's revolution is almost equal to year and position, longitude and latitude of sun depend on date of calender with minor corrections. * 116 I Siddhdnta Darpand * Moon's orbit has perturbations due to attract tion of sun and other planets. Movement of its node is faster (Due to its nearness to earth and effect of sun, parallax etc, its accurate calculations for eclipse is needed. First, we derive the formula for calculation. To know the true position - (1) A planet mi its etiptical orbit with sun at focus is calculated to know its direction and distance from sun (Heliocentric position) (2) Position of earth is calculated from sun.? From its direction and distance we calculate! Figure 3 direction and distance of planet from earth (Geocentric position). Explaination of Anomalis (Fig 3) - APA' is eliptical orbit of a planet ^nd S is the attracting sun at a focus. Revolution of the planet is counted from position A when it is closest to sun (near end of major axis) After 'd' days planet is seen at position P, L ASP is manda kendra = (True anomali). Auxiliary circle is drawn on diameter AA'. PBi is perpendicular on major axis, on extension itj meets anxiliary circle on P\ ZAOP' = <p is called eccentric anomaly. 6 and <p are measured in length^ of arcs (radian measure). If daily mean motion of planet is n radians, then 2ji/u is the period of „, *c 117 True Janets Virion If daily motion is always n, then the SS after 'd' days with be 'nd' which is called P pan anomaly (madhyama manda kendra). This Sffl be true anomaly, if speed of planet (angular) Tmnstant. According to second law of Keplar -- lS _ AreaASP = d = d , 2* = dn Area of ellipse Time of revolution n 2jt Area ASP _ area of ellipse b Jtab Area ASF " area of circle a ^ a 2 where a and b are semi major and semi minor axis. Area ASP _ Area ASP = Area ASP SO Area of ellipse ~ Area of circle jra 2 But Area ASP' = Area AOP' - Area, SOP' a 2 <I> BP x OS _ a 2 ^ _ asinOae = T "■ 2 " 2 2 a * = — (0 — esinO ) Where e = eccentricity (cyuti) of ellipse. dn _ Area ASP _ a^ (<f> - esin<I>) Hence — = Me& of eUipse 2 ^2 or dn = <X>-e sin <I> (1) This is relation between mean anomaly and eccentric anomaly. Relation between true anomaly and eccentric anomaly-Polar equation of ellipse is a (1 - e 2 ) 1 + ecosB As per difinition of ellipse SP =e x distance of P from directrix =e x distance fo B from directrix (1 to major axis) 118 Siddhdnta Darpana =e (distance from centre to directrix - centre to B) = e (J - OB) a = e(- - a cos <X>) = a - e a cos O e or, radius vector (karna) = a (1 - e cos <p) — (2) a (1 - e 2 ) So = a (1 - ecos^>) 1 + e cos 6 / Or, 1 + e cos = 1-e 2 1. - e cos <b a 1 ~ e -, (e cos <p - e) or, e cos 6 = — - 1 = - — ~ L 1 — e cos <p 1-e cos <I> cos <I> — e or, cos 6 — 1-e cos <£ _ 2 & 1 ~ cos # 1 - ecos <I> - cos <I> + e Tan — = — = — 2 1 + cost? 1-e cos <I> + cos 3> — e _ 1 + e 1 - cos<l> 1 - e ' 1 + cos 1 + e 2 O = _ f an 1-e 2 -v/ i+7 $ or tan- = V . tan -~- (3) 2 1 — e ^ Equations (1), (2) and (3) can be used to find manda kendra (True anomaly), manda karna (distance from sun) and d (time in days) from A (perihelion-nearest point). For practical purpose, these equations are not convenient. For calculation on basis of average velocities which are known accurately, equation (3) needs to be expanded in a power series of small e as coefficient of sines of average position. 119 Tru e PI*** Equation (3) can be re-written on basis of formula of Trigonometry tane = -^ x e + e -i j ' where e = base of natural logarithm 1 11 = 1 + -[1 + |2 + 13 + - To differentiate it from eccentritity, we write is E then (3) becomes _.^ i(E2 + E 2 ' i( E 2 -i* E* - 1 _ t/ltc E" - 1 01 E 1 * + 1 ~ 1 ~ e * E i4> + 1 Adding 1 to each side and substracting from 1, then dividing vr^r (E i4> + i) - vrr^ (e 1 * " *) .„ vr+ r a?* - i) + vr^ ce^ + i) E i<D _ vT~T~e - vT~^"e = : dk where p = TrvTTlr^ 1 - pE or E = E ^^ >i* 1 - pF Taking logarithm of both sides, i<9 = i<D + log (1 - pE _i *) - ^g (1 -pE 1 "') + 2! (E 3i^» _ E -3i* } + .... 3 v 120 Siddhanta Darpana or 6 = d> + O + tCE 1 * - E- i4> ) + E_ (E 2i4> E" 2i *) + £? (E 3i* _ E - 3i V 3i or 2p 2 2p 3 = $ + 2p sin <D + -£- sin2 <D + -*j- sin 3 3> + . . . P 2 P 3 or ^ = +2 (p sin 0> + y sin2 0> + *- sin3 . . . ) (4) Equation (4) needs to be expressed as a series in mean velocity n and d which are easily determined. For this, we use Taylor's infinite series based on Lagranges mean value theorem of differential calculus. This is written as h 2 f (x + h) = f (x) + h f (x) + — f" (3^ + .... oo let y = x + h f (y) (5) Then F(y) = F [x+h f(y)]. Then by Taylor's Theorem, h 2 d = F(x) + h.f(x).F (x) + -j^ Wy)-F (x)] [3'dx 2 [f(y) 3 .F (x)] + ... + S £^ ^ ' (X)1 + ^ From equation (1), dn - <S> — e sin O or <I> = dn + e sin <I> = m + e sin <J> where m = dn This is in form of (5) where y « O, x=m, h = e Let F (O,) * <t>„ then F(m) = m and F (m) = 1 True Planets Hence form (6) . e 2 <* 2 <j) = M + esinm.H — 33 (sin m [2 dm 1) + L 3 ' dm <r 3 (sin m.l) 121 + d 3 4 (sin .m .1) + e 5 d 4 — (sin m .1) + . . . L 4 ' dn 3 |5 dm' Expansion of Sin n m is given by (when n is even) • 1 — — : — [ cos nm - n cos (n — 2) m 2 n_1 (-if 2 n (n - 1) n (n - 1) (n - Z) + .- — - cos (n - 4) m - — - — - ' sin 11 m = 12 cos (n — 6) m + . . ■ . ] When n is odd, then 1 |3 sin n m = n -1 2 n_1 (n - 1) — r *• . n (n — 1) [sin nm - n sin (n - 2) m + — ^— — - sin (n - 4) m n (n - 1) (n - 2 ) — — sin (n - 6 ) m + ......]. Hence - — (sin m ) = d (1 — cos2 m) dm dm 2 d 2 (3 sin m - sin3m^ = sin 2 m 2 ( sin m ) = 2 dm z dm 2 = d (3 cos m — 3 cos 3 m dm — -7 (3 sin 3m - sinm) 4 222 Siddhdnta Darpana d 3 , . 4 , d 3 ^ (Smm) = ^ 1 2x3^ (cos 4m - 4cos 2m + — ^— x 1) 2 3 (-1) 2 L2 - (4 3 sin 4m-4 x 2 3 sin 2m) = 4 (2 sin 4m - sin 2m) d 4 dm (sin m) dm' 1 5x4. ^ (sin 5m - 5 sin3m + — — sin m) 2 4 (-1) » = — (5 4 sin 5m - 5 X 3 4 sin 3m + 10 sin m) 16 v dm" 6 * (sin 6 m) = dm 5 2 5 (-!)" (cos 6m - 6 cos 4m 6x5 „ 6 x 5 x 4 + .. cos 2m ^ x 1) |2 L3 = — (6 5 sin 6m - 6 x 4 5 sin 4m + 15 x 2 5 sin 2m) 32 v Hence <J> = m + e sin m e 2 e 3 3 „ . . _ . , + - sin 2m + — x 7 (3 sm 3m - sin m) 2 L 3 4 e 4 e 5 1 + — x 4 (2 sin 4m - sin 2m) + yr x — x (5 4 sin 5m - 5 x 3* sin 3m + 10 sin m) + — . — (6 5 sin6m - 6 x 4 5 sin 4m + 15 x2 5 sin 2m) + . . Separating sin m, sin 2 m ...etc. x J- o 4 e e <t> = m + (e - - + j£) sinm + {- - - + ^) sin 2m + ( 3e 3 27e 6 48' 4 ,._6 8 i28 )sin3m + (|-^-)sin4m True Planets 123 125e 5 + -rrr~ sin 5m + . . . (7) Next quantities contains powers of e or more hence they are very small and left out (e is very small because orbit is almost circular with very small eccentricity) Equation (1) can be also written as e sin O = <S>-m <I> - m or sin <l> = e From (7), this becomes 2 e 4 e 3 e 5 sin <D = (1 - — 4- — ) sin m + (- - — + ^) sin 2m 2 „_4 / e 4e 5N e* 27c . m + (3 T"W sin3m + 125e 4 15 \ sin 4m H — — — - sin 5m + . . . . (8) 384 Now expansion of Sin 2 O, sin 3 O — are to be obtained Now in equation (6), take F (<I>) = sin 2 <!>, then F(m) = Sin 2m and F' (m) = 2 sin 2 m Hence equation (6) becomes - sin 2 <p = sin 2m + e sinm x 2 cos 2m e 2 d . 2 . . e 3 + Tr - — (sin m x 2cos 2m) + |_2 dm v J L 3 12 e 4 d 3 (sin m x 2 cos 2m) + .2 v J 1 4. j_3 9 dm 2 L 4 dm : 224 Siddhdnta Darpana ♦4 (sin m x 2 cos 2m) + e 5 d 4 ■_5 L^dm' (sin m x 2 cos 2m) + ... In this, sin m x 2 cos 2m = sin 3m - sinm 1 - cos 2m •_2 - — (sin m X 2 cos 2m) = - — dm dm x 2 cos 2m \ dm (cos 2m - cos 2m) = dm cos 2m — 1 + cos 4n0 V = 2 sin 4m - 2 sin 2m d 2 dm : (sin 3 m x 2 cos 2m) dm' [3 sin m — sin 3m \ 4 x 2 cos 2m f 3 sin m cos2m - sin 3m cos2m" dm J / dm j dm' 3 1 - (sin 3m - sin m) - - (sin 5m + sin m) 4 4 — (3 sin 3m - 4sin m - sin 5m 1 -y i = - (- 3 sin 3m + 4 sin m + 5 sin 5m) 4 •4 dm' (sin m x 2 cos 2m) = dm' — (cos 4m — 4 cos 2m + 3) 2 cos 2m 8 -5 - (2 cos 4m . cos 2m - 4 x 2 . cos 2m dm 3 18 V + 6 cos 2m) True Planets dm* - (cos 6m + cos 2m) 8 v 125 4 6 — (1 + cos 4m) + — cos2m 8 8 dm' - (cos 6m - 4 cos 4m + 7 cos 2m - 4) 8 = i (6 3 sin 6m - 4 4 sin 4m + 7 x 2 3 sin 2m) 8 v so {sin2 O = sin 2m + e (sin 3m - sin m) +— (2 sin 4m - 2 sin 2m) e 3 1 + — - (25 sin 5m - 27 sin 3m + 4 sin m) [3 4 V e 4 1 + — . - (216 sin 6m - 256 sin 4m + 56 sin2m) + 13 8 e 3 2 7e 4 = (- e + — ) sin m + (1 - e z + — ) sin 2m 9e 3 ? 4e 4 25e 3 +(e- — ) sin3m+ (e z - — ) sin4m + -^- sin5m + .... 8 In equation (6), now take F(0)= sin 3<£, then F(m) = sin 3m and F(m) = 3 cos 3 m, then it becomes - sin 3 O = sin 3m + e sin m x 3 cos 3m + 2 [2* dm d 2 (sin m x 3 cos 3m) e 3 d 2 f + -7-r . r | sin 3 m x 3 cos 3m + ...K [3 'da. 2 4 3e = {sin 3m + — (sin 4m - sin 2m) e 2 3 + — . - {(5 sin 5m - 6 sin 3m + sin m) 3 4 v 9 226 Siddhanta Darpana + or, t 1 6 ' 8 (36 sin 6m - 48 sin 4m + 12 sin 2m) + ... sin 3 <J> = 3e; 8 sinm 3e 3e' (y- 4 9e z ) sin 2m + (1 — — — ) sin 3m 3e 2 15e z 9e 3 (— - 3e ) sin 4m H — - sin 5m + — — sin 6m + .... Similarly sin 4 <J> = sin 4 m + e sin m x 4 cos 4m e 2 d + — . -r- (sin 2m X 4 cos4m) + — 2 dm ' = Sin 4m + 2e (sin 5m - sin 3m) + — (6 sin 6m - 8 sin 4 m + 2 sin 2m) + 2 2 or sin 40 = e sin 2m - 2esin 3m + (l-4e ) sin 4m + 2e sin 5m + Sin 5 <I> = Sin 5 m + — e (Sin 6m- Sin 4 m) + — 5e 5e = — — - sin 4 m+ sin 5m + — sin 6 m+ — 2 2 Value of p can be known in terms of e by expanding with binomial theorem also (Taylor's theorem is not needed) _ VI + e - vT^~e P ~ Vl + e + VI - e 1 - VT"^" 1 2 /2 e .6 1 1 e e e e ( T + ¥ + 16 + "") or P = 2 + T + ^ + - P 2 = / 3 5\ 2 e e e 4 + 8 + 64 True Planets 12? 3 fe e 3 e 5x e 3 3e 5 9e 7 ' = ~8~ + 32 + 128 + V e 4 . 5^ 4 + 8 + 64 'e 2 e 4 5e 6 — + — + — 4 8 64 16 + 16 + / .5 _ e 16 + 16 3 5\ £ £1 £1 2 + ~8~ + 16 e = 32 + Now equation (4) can be written as = m + sin 2m + 3 5\ e e e ~ ~8~ + 192 V 3e 3 27e~ :\ sin m + 8 128 \ / ■\ sin 3m + / £l_ £ £! I 2 " 6 + 48 e ~3 15 / 125e 3 sin 4m + — — — sin 5m + . . . . 3 + 2 { ( f + y + i6 ) [ (1 ~T + ^ )sinm e e 2 e 5 + ( 2-T + 5 )sin2m + ( 3e 2 27e- 4e- 8 118 ) sin 3m + (— - — - ) 15 125e :> sin 4m + sin 5m + .... ] 1 e 2 e 4 5e 6 2 7e 4 sin m + (1 - e + — (-e + j) ) sin 2m 9e 4e 25 e +(e - — -) sin3m+(e 2 - — r-) sin4m H — — — sin5m +..] 1 + - 2 'e 3 3/ 32 \ / _ 3e' 8 3e 3e 3 « sin m — (— - — - ) sin 2m 128 Siddhanta Darpan 9e 2 + (1 T~ ) sin 3m (3e 5 15e 2 9e 2 + ~Y ~ ^ ) sin4m+ — sin 5m + — sin 6m +....J 1 ~ 4 6 l e e + 4 ^16 + 16^ C sin2m ~ 2e sin3m + (1 + 4e 2 ) sin 4m + 2e sin 5m] 1^. 5e 5e 5 32 *~ T Sm 4m + sin 5m + Y sin 6m ^ Terms beyond e 6 and sin 6 m have been left out as they are neglible. Collecting the multiples of sin m, sin 2m —etc. e=m+(2e _^ + ^ )sinm +( ^_m + i^ 4 96 ^4 24 192 o- o , / 13e3 43e5 103e 4 451e 6 sm 2m + (— - — - ) sin 3m + ( i^_ __ ^_ . 12 64 v % 48Q ) . A 1097e 5 Sm ~960~ Sin 5m (9) Actual equation for knowing heliocentric true position - Equation (9) is the main equation from which helocentric position of planets are calculated from their mean speeds and eccentricity of orbits. This is called manda karna in Indian system. For example, in case of Jupiter, e = 0.048254, hence e 2 = 0.0023284, e 3 = 0.0001124, e 4 = 0.0000054, e 5 and higher powers are very small and can be neglected for calculation of 1" accuracy. For Jupiter - = m + (0.0 96508- 0.0000281) sin m + (0.002 9 106 - 0.0000025) sin 2m True Planets J29 + 0.0001218 sin 3m + 0.0000058 sin 4m + - or 6 = m % + 0.0964799 sin m + 0.002 9081 sin 2m + 0.0001218 sin 3m + 0.0000058 sin 4 m + — (10) If the sines are expressed in kala or vikala in Indian system, then the value of 6 will come in kala or vikala and this will be manda phala of guru from centre of sun. If they are expressed in fractions, the terms after m will be in radian. To convert them in kala or vikala, they are to be multiplied by 3437.75 or 206265. Equation for any planet can be obtained by putting its eccentricity e in equation (9) The eccntricities are given in end of this section. Helocentric distance - Manda karna (Heliocentric distance of planet) SP = a(l-e cos O) Putting F(*) = 1 - e cos 4>, F(m) = 1-e cosm, eiv? = 6 Sm m ' in e< l uation &)> ^ylor's series gives 1-e cos <& = (l-e cos m) + e sin m — (l- dm cosm)+ L3 dm f — (Sin 2 m x e sin m 1 3 • ^5 ( Sin m x e sin m) 130 Siddhanta Darpana f 1 = 1-e cos m + — - — cos2 m + — cos 3m - — cos j 4m + — cos 2m 3 i e 2 3e 2 e 2 „ 2e 2 „ i = (1 + y ) - e (1 - — ) cos m - y (1 - — ) cos 2m : 3e 3 cos 3m + 8 Hence, radius (karna) e 2 3e 2 e 2 2e 2 x = a [ a + y ) - e ( x - -j- ) cos m - y ( a " X } 3e 3 cos 2m - — cos 3m] (11) Semi major axis (smallest+largest distance),/ 2 of Jupiter a is 5202.8 hencee equation of its redius is 5202.8 [ (1+0.0011642) - (0.048254 - 0.0000421) cosm - (0.0011642 - 0.0000018) cos 2m - 0.0000421 cos 3m)] = 5202.8 (1.0011642 - 0.00482119 cos m - . 0.0011624 cos 2m - 0.0000421 cos 3m) = 5208.86-251.06 cosm - 6.05 cos 2m - 0.22 cos 3m Semi major axis has been expressed as ratio of earth's mean distance from Sun which is taken as 1000 Parameters of planetary orbit Constants for earth - a© = 1.4959787xl0 n metres, e is symbol for earth, a is semi major axis Time period of revolution T e = 3.1558150 x 10 7 sec. 737 True planets Mass m© = 5.976 x 10 24 kg, Moment M© = 2 6 6 x 10 40 kg m 2 /sec Eccentricity e© = 0.0167 Orbits of other planets Planet a in Peri- Mass Mome- inclinati- Eccentric- a© od (in m©) nt on of ity e Years (in m©) orbit Mercury 0.38 0.24 5.6 x 3*x V 0' 14" 0.2056 71 10 10 Venus Mars Jupiter Saturn Uranus 0.72 0.62 8.1 x 7.0 x 3° 23' 39" 0.0068 33 10 -1 10 -t 1.52 1.88 1.1 x 1.3 x 1'51' 0" 0.0934 37 10- 1 10" 1 5.20 11.87 3.2 x 10 2 7.6xl0 2 1° 18' 21" 0.0484 28 9.53 29.46 9.5 x lO^.&riO 2 2'29"25" 0.0557 89 19.18 84.01 1.5 x lO^^xlO 1 0°46" 23" 0.0472 Neptune 30.06 164.8 1.7X10 1 9.5X10 1 1°46' 28" 0.0086 Pluto 39.44 247.6 2.0 x 1.2 x 17° 8' 38" 0.2486 10' 3 lO' 2 Conversion of Orbital distance to ecliptic distance Equation (10) gives the distance (angular) of planet in its orbit from its nicha (perihelion) or closest position. If the orbit of planet would have been in same plane as earth's orbit (or plane of A' P' Figure 4 - Inclination of orbit with elliptic 132 Siddhdnta Darpana ecliptic), this would have been its distance in ecliptic also. But every planet's orbit is at an angle with ecliptic which is its parama sara (maximum distance from ecliptic). This inclination is given in the chart above. There is no inclination for earth's orbit (or sun) because it is measured from this orbit only. In fig 4, PC is orbital ellipse and CP' is the ecliptic. S is centre of sun and A is perihelion (nica) of the planet. P is true position. PP' is perpendicular on ecliptic, hence it passes through pole of ecliptic. Then ASP is orbital true anomaly (Kaksa spasta kendra) and SP is spasta karna. AA' is perpendicular on ecliptic and also passes through its pole. Distance A'P' along ecliptic is the ecliptic true anomaly (kranti vrttiya spasta kendra). For theoretical calculation, it is easier to find out relation between CA and CA' or CP and CP'. But in practice, we need to know only the minor correction to orbital distance to know ecliptic distance. This correction or difference between orbital distances from pata C (intersection point of orbit and ecliptic) is called parinati. Nica parinati = CA - CA' Planet parinati = PA-P'A PP' is instantaneous or istakalika sara, Z.PCP' is parama sara (equal to maximum angular distance from ecliptic), PC is distance from pata to graha or vipata graha. Z.PP'C is right angle, hence PCP' is a spherical right angle triangle. From Napier's laws - True Planets 133 (1) Sin (90° - CP) - cos (PP') x cos CP' (2) Sin PP = cos (90 Q -PCP) x cos (90° -CP) (3) Tan PP' = Sin CP x tan PCP' (4) Tan CP' = cos (PCP') tan CP Sin (CP-CP) = Sin CP. cos CP - cos CP. sin CP -(12) From formula (3), tanPP sin CP ' = tanPCF ¥ sin CP Formula (4), - = cos (PCP ) tan CP cos CP __' sin CP •. cos CP = cos (PCP) tanCP tanPP cos CP { = ; x tanPCP cos PCP sin CP tanPP cos CP x sin PCP sin CP so, sin (CP - CP ) = sin CP -^ x ^g - cosCP sinPCF sinCP tanPP tanPCP tan PP x cos CP cosCP x tanPP sin PCP tanPCP tanPP x cosCP sinPCP [1 - cos PCP J __ sinPP cosCP : x 7 x vers sin PCP cosPP sinPCP From formula (2), — - t = sinCP sin PCP 234 Siddhanta Darpana Hence sin (CP-CF) sin CP x cosCP cos PP r x vers sin PCP Parama sara of all planets except Budha is less than 3.4° hence their istakalika sara will be still smaller. Hence Cos PP' s 1. Then Sin (CP-CP') = Sin CP cos CP x vers sin PCP' = 1/2 sin 2 CP. V sin PCP', or, sin (Parinati) = 1/2 X versed sin of parama sara X Sin (2 x vipata graha) — (13) Equation (13) gives correction to find position of planet in kranti vrtta. Geocentric position Figure 5 - Geocentric position of planets To find the direction and distance of planet from earth, we have to know the position of earth itself. Position of earth also can be known from equation (9) like other planets. Position of Sun from earth is opposite to earth from sun direction i.e. 180° away. Sighra kendra is difference of ecliptic spasta kendra and position of sun from earth. TrU e Planets 135 In Figure 5, S, E and J are positions of sun, ar th and Jupiter. ESS' is direction of Sun from 6 a rth (both centres). S' is its position in ecliptic. c'Sj is sighra kendra of Jupiter. ZESJ = 180°-S'SJ and in AEJS, two sides ES, SJ and angle between them is known. Then EJ, L SEJ and ^EJS also can he known. From trigonometry ^ «FJ-STE ST-SE SEJ + SJE tan 2 = SJ + SE tan 2 Here L$ EJ+^SJE=ZS'SJ = sighra kendra SEJ-SJE SJ-SE sighra Kendra /.tan — — " S J + SE tan 2 From this difference of angles Z.SEJ and ASJE can be known. Their sum (sighra kendra)) is already known. By adding these and dividing by 2 we get ASEJ which is angle between Jupiter and Sun as seen from earth. This is called Inantara (Ina=Sun). Distance of Jupiter from Earth JE is sighra karna. TE JS fcI . Lz. - - — : by sm ratios Sin L E S J sin L S E J But sin ^ISEJ = Sin (180°-^SEJ) * ^JSS' = Sin (Sighra kendra) Figure 6 - Sighra Kendra in ecliptic 236 Siddhdnta Darpana Hence, Sighra karna JE = Sin of SIghra kendra x manda karna x - — _ — - —(14) sin (inantara) ■ ■ ■ .' In fig 6, XX' is ecliptic plane which contains earth's orbit EYZ. Orbit of Jupiter is CJC which cuts ecliptic on C and C. C is north pata and C south pata. S,E and J are true positions of Sun, earth and jupiter.*- JJ' is perpendicular on ecliptic plane. V is point of vernal equinox (north pata of ecliptic and equator planes). ZJSJ'is heliocentric inclination of Jupiter, Z.VS]' is longitude of planet (angle in ecliptic plane between vernal equinox and planet - seen from sun. JEJ' and V'EJ' are geocentric inclination and longitude of jupiter. SVMEV. ^V£E is heliocentric longitude of earth, hence ZVSE + 180° is geocentric longitude of sun. (A SEJ' is same as A SEJ of figure 5) SIghra karna and inantara in equator are EJ' and Z.SEJ'. True sighra karna and inantara are EJ and Z.SEJ; EJ = cos^JEJ- ~< 15 > L J E J' is very small, hence is cosine is almost 1. This is only a rough outline of calculation of palnetary positions in modern astronomy. There are prturbations in positions of earth due to effect of moon, jupiter and venus (others negligible) Similarly prerturbations occur in other planets also for which corrections are necessary. There is slight change in eccentricity and positions of pata also which cause other corrections. The corrections to the orbit of moon are more important because it has largest effect on earth's tides, climates, calender and eclipse etc. frue Planets 137 (3) Tables of Sun - Precession of equinoxes - According to New- comb, rate of general precession in longitude per tropical year of 365.2422 days is 50". 2564+0". 02223 (t/100) + 0".0000026 (t/100) 2 where t is measured in tropical years from 1900.0 AD. Annual rates of precession per sidereal year f 365.25636 days is 50".258 35 + 0". 02223 (t/100) + 0". 00000026 (t/100) 2 In Julian year of 365.25 days, precession is 50."25747+0".02223 (t/100) + 0".0000026 (t/100) 2 In Indian system, initial point from which longitude is measured is a fixed point of ecliptic with respect to stars. In modern astronomy, it is the point of vernal equinox. Distance from fixed initial point of vernal equinox point is called Ayanamsa. To fix initial point accurately, star spica (avirginis) has been assigned a nirayana (from fixed point) longitude of 180°. Since the star also has some small motion, its longitude of epoch time is taken when fixed point and vernal equinox point were together with sun on it. This epoch of 0° Ayanamsa (0° sun also) was on 285 AD, March 22, 17h 48m E.T. or 21h 27m 1ST. That was beginning of saka era 207, Samvat era 342 and Kaliyuga era 3386. Julian day on March 22 noon was 1825325 and kali elapsed days at midnight was 1236770. The day was Sunday. Mean su n (both tropical and sidereal) was 0°0'0" and (Mean moon - Mean sun) = 351 '.67. Thus it was also a new moon day. In Besselian fictitious year, epoch was 138 Siddhdnta Darpana 285 79.994 360 285.2222 A.D. The epoch is 1614.7778 years before 1900.0 AD. From 0° Ayanamsa of this epoch to ayanamsa at 1900 AD, January 0.813 i.e. 19h 31m ET is 22°27'43".51. Thus Ayanamsa from 1900.0 AD is ' t x 100 A = 22°27'43". 51+50". 2564t+l". 1115 3 't^ 2 100 \ / +0".0001 This is formula in tropical years. In sidereal years, it is A = 22°27'43."40+50".25835t+l".1115 't* 2 / t v3- 100 + cr.oooi 100 \ t In Julian year, formula is A = 22°27 / 43". 40+50". 2575t+l". 1115 3 't* 2 100 100 \ / + 0".0001 Daily rate of precession in 1900 AD = 0". 137597 If time is taken from 285AD epoch, formula in tropical years is A=49".8981t+l".1073 (t.100) 2 + 0.0001 (t/100) 3 + — There are similar formula for sideral and Julian years. Position of star Spica of 180* long in 285 AD - In 1950.0 AD its position was 139 True Planets R.A. - 200° 38' 19". 6, Declination = -10'54'3".4 Annual proper motion A« = -0".039 A<5 = _0".033 Tropical longitude = 203° 8'36"3 latitude -= -2°3'2".8 Sidereal long = 179°.58'59".7 (Ayanamsa 23°9'36".6) Annual proper motion in ecliptic system is *AA = - 0".0232, A0 = -0". 0449 Due to slow motion of plane of ecliptic, longitudes and latitudes of fixed stars undergo changes. Annual rates are as follows - AA=7rcos(A-n)tA n £ A /3 = - * sin (A - II) In 1950 AD I* 285 AD [1 = 0".4708 0^.4824 A=203° .4', p = -2° 3', A = 180° 0' ,fi = - 1'56' [1 = 174°24' (Trop) 159° 12' Hence AA = - 0". 0147 - 0". 0151 A p = -0" . 2283 -o" • 1713 Average value of AA= -0".0149, A0 = - 0".1998 Proper motion - 0".0232, - 0".00449 Total anual variation AA= -0".0381, A/3 = -0".2447 In 1665 years (1950-285 AD), total variation in longitude is - 63". 4, in latitude - 6'47".4. Then nirayana longitude in 285.22 AD is 180°0'3".l and latitude is - 1°56'15".4 Thus at epoch, ^its nirayana longitude was 180° approx. '?■# 40 Siddhanta Darpana Obliquity of Ecliptic to the equator - E = 23°27'8".26-46". 845T - 0".0060T 2 +0".001837T 3 where T = Julian century of 36525 days from 1900.0AD E.T. Rate of variation per century is de — = - 46". 845 - 0".012 T + 0". 00549 T 2 When T 3 term has appreciable value, century figures need some correction. Then putting T = Tc+ I5o (Tc = completed centuries, t = extra years) e = 23'27'8".26-46" 8457 Tc - 0".006T 2 + 0.00183TC 3 + (-0".00651 Tc + 0".00549 T c 2 ) x — ; 100 Mean Longitude of Sun - (L) - Epoch is 1900 AD, Jan 0.0 ET. i.e. 0h0'4".4 universal time, T = Julian centuries of 3 6525 ephemeris days from epoch. According to Newcomb, sun's mean tropical longitude, freed from aberrations is L=279'12 / 13 // .88 + 129602768 7 '. 13 T+l".089 T 2 Motion in a century of 36525 ephemeris day is 12960 2768/'13 = 360° x 100+27".6813 x 100 = + 46'8 /7 .13 Daily motion is 0°59 / 8 ,, .3304074 If Tc is completed century, t = remaining years, d = extra days, True Planets 142 L = 277°!2'13".88 + (46'8".13» Tc + (59'8".330) d + 1".089 Tc 2 + 2.178 Tc x ^ Sidereal or Nirayana Mean Sun (L') is V = 256°44'30".48 + 129597742". 38T-0".0225T 2 4)."0001T 3 Motion in a century = 360° x 100-22".5762 x 100 = -37'37".62 Daily motion = 35488". 192 80988 = 0°59'8".1928098 Sun's Perigee (=11) and Mean anomaly (=g)) Trop II = 281°13'14".92 + 6189".03 T+1".63T 2 + 0".012T 3 Sid IF = 258°45'31".52 + 1163". 28 T + 0."52 T 2 + 0".012T 3 Motion of IF per century = 19'23".28, per year = 11". 63, per day = 0".0318 Mean anomaly of the earth or the sun = g = L - II or L' - IF g = 357°58'58".96 + 129596579." 10 T-0".541T 2 - 0".012T 3 Daily motion = 0°. 9856002670 = 3548". 160961 Hence the period = 365.2596413 ephemeris days. Mean anomaly M in days is obtained by dividing g by daily motion and adding a constant of 5.37018 days M = 3.12376 + 36525 T - 0.0001525T 2 " °- 0000034 1 ; * 242 Siddhdnta Darpana 36525 days = Period x 100 - 0.96413 days. Mean Elongation of the Moon in days Brown's Moon = 263°50'45".48+1732564379".31T - 4".08T 2 +' 0".0068T 3 Newcomb's sun = 279° 12'13".88 + 129602768". 13T + 1".089T 2 D=Moon - sun = 344°38'31".60+1602961611".18T - 5".169T 2 + 0".0068 T 3 Daily motion of D = 43886" . 697089 Period = 29.53058867 days Converting into days D = 28.27079 +. (period x 1236+25.192399) T - 0.0001178T 2 + 0.000000155T 3 Venus and Sun - Mean Tropical Venus is 341°57'57".49 + 210669162". 88T + 1M148T 2 V = Venus - Sun = 62 °45'43". 61 +81066394. "75T + 0".0258T 2 Daily motion of V = 2219". 4769, period = 583.921373 days. Converting into days V = 101.8004 + (period x 62 + 321.87487) T + 0.0000116T 2 Sun and Jupiter - Mean Tropical Jupiter is 238 o 0'27".69 + 10930687. 15T + 1".205T 2 J = Sun - Jupiter = 41° 11'46".19 + 118672080".98T - 0M16 T 2 Daily motion of J = 3249". 064503 Period = 398.884048 days Convereting into days, we get j = 45.6458 + (Period x 91 + 226.55163) T 0.000036T 2 Nodes of Moon - Tropical longitude of the node is Q = 259 12'35".11-6926911".23T+7".48T 2 + 0".008T 3 - Q = 100°47'24".89+6926911". 23T-7",48T 2 _ 0".008T 3 Daily motion = 190". 63412, Period = 6798.36327 days converting into days expression for - Q and adding a constant of 0.818 days, No. of days N since tropical longitude of moon's mean node was zero is N = 1904.177+ (period x 5 +■ 2533.1835) T - 0.003924T 2 - 0.000042T 3 Julian day Number : Pope Gregory introduced in 1582 AD year of 365.2425 days by omitting 10 days (Oct. 5 to Get. 14)) from calender. Before that, there was leap year in every 4 years. In Gregorian calender, 97 leap years come in 400 years. Years divisible by 4 or centuries by 400 are leap yeears of 366 days. Normal year is of 365 days. Julian days are numbered serially from Jan 1, 47 13 B.C., Monday at Greenwitch mean noon. Besselian Fictitious year begins when the topical mean sun is 280°0'20".5 or the same unaffected by aberrations is 280 °0'0". Notation like ^00.0 AD. is used for this year. 144 Siddhdnta Darpatut Let K = time from beginning of Besselian year upto beginning of calender year i.e. Jan 0, oh E.T. for common year or Jan 1, Oh E.T. for leap year. Day from beginning of ficitious year = Day of, year + K K = - 0°48'6".6+129 602768M3T + 1".089 T 2 Daily motion = 3548" 3304074 Period of length of Tropical solar year = 365.24219878 days K in days = - 0.8135 + (perod x 100 + 0.780122) T + 0.000307T 2 Inequalities of long period in mean longitude <5L=+6".40 sin (231°. 19+20°. 20T)+(1".882- 0".016T) x sin (57°24+150\27T)+0".266 sin (31°. 8+119 °.0T)+0".202 sin (315\6+893°.3T) First term has a period of 1782.2 years (century variation of 20°. 2) i.e. 1° in 3548 days. <5L = + 6".40 sin [(AD year - 755.5)x0.202)] Equation of Centre: e = eccentricity of orbit/ g = mean anomaly (written as m in derivation of formula). Equation of centre e 3 5 2 He 4 13 % (2e - -) sin g + (-e 2 - — ) sin 2g + — e 3 . „ 103e 4 sin 3g + -^psin4g Here, e = 0.016,751, 04-0.000,041,80T- 0.000,000,126T 2 = 0.016, 75104-0.000,041,80 (T+.00 301T 2 ) Multiplying by 206264.8 we get e = 3455". 150 - 8".621 (T+0.003T 2 ) True Planets 145 So equation of centre = + 6910 / .057 Sin g + 72'.338 sin 2 g + 1".054 sin 3 g + 0".018 sin 4g •- 17".240 (T + 0.003T 2 ) sin g - 0".361T sin2g. perturbations to Sun - Action of Moon - Longitude of sun (or earth in opposite direction) is the longitude from centre of mass of earth and the moon. This is called geometric longitude. The origin is to be transferred to centre of earth. Radius of earth is taken as unity, f]' an d 17 are horizontal parallaxes in seconds of arc of moon and sun respectively, /}' are /? are their latitudes. Distance of mass centre from earth centre in direction of moon 206265 _ 2506.3 " 82.30 ft' I! ' (Ratio of earth mass to moon mass is 81.30 adopted in 1968) AL = 2506.3 x ^ Cos£' sin p-O) P = 2506.3 x Jl Sin p Substituting numerical values - AL = + 6". 44 sin D - 0".42 sin (D-g') Sun's latitude /J = + 0",58 sin U, or + 6". 44 sin fi' or roughly 0.11 x moon's latitude in seconds U = Mean moon - lunar node A log R = + 0.0000134 Cos D. Action of other planets Action is calculated in terms of Q = difference m heliocentric latitudes of the planet and earth. 146 Siddhanta Darpant^ Due to elliptical shape the deviation due to planets also depends on g and (W'-W) where g = mean anonaly of earth (i.e. of sun) j W = Longitudes of the planet's perihelion, | W = perihelion of earth J T is in 100 years from 1850 AD then | K' = W'-W = 29 •5'55 ,, -18 / 40 ,/ T (Venus) K" = W"-W = 232°56'11" + 7'18"T (Mars) K "' > W w -W = 271°33'16" - 6'33"T (Jupiter); Century variations of these quantities are very small, and they dm be 'considered as constants fbfcj 1000 years or more g = hehiocentric lat. of earth - W g' = hel. long, of the planet - W (e.g. for :\ venus)) , = planet - k'-W = (Planet-earth) + (earth - W) - K' = Q + g - K' Perturbations due to venus (Approx New- comb formula) Pert = + 4.84 sin Q - 5.53 sin 2 Q-0.67 sin3Q - 0.21 sin 4Q - 0.12 sin (2Q+g) - 2.50 ski (g+12°-2Q) - 1.56 sin (g+12°-3Q) + 0.14 sin (g+12°-4Q) - 1.02 sin (2 g + 40°-3Q) - 0.15 sin (2 g+40°-4Q| +0.12 sin (2g+40 # -5Q) - 0.15 sin (3g+56°-5C$ Corresponding formula given by Le-Verrier fej| True ^^ s U7 Pert = + 4.91 sin Q-5.61 sin 2 Q-0.67 sin 3Q 21 sin 4 Q - 2.52 sin (g-2Q + W-90°) - 1.58 sin (g-3Q+w-9(T) For first approximation, calculation is based on Q only, then for M = g+5\29. it is calculated Perturbations due to Jupiter - Newcomb formula is Pert = + 7.21 sin (Q-l°5')-2.73 sin (2 Q - 0°15') - 0.16 sin (3 Q + 4'5r) + 2.60 sin (Q+g-84'46') - 1.61 sin (2 Q + g - 22°. 6) - 0.56 sin (3 Q + g + 87°2) - 0.16 sin (g-Q+20'.l) - 0.21 sin (3 Q + 2 g + 77°) First three terms according to Le verrier are + 7/20 sin (Q-l°5') - 2.73 sin (2 Q-18') - 0.16 sin (3Q + 5°) These terms are tabulated for Q, then for Q and M. Perturbations due to Mars - Newcomb for- mula is Pert = + 2.04 sin (2Q+15') + 0.27 sin (Q~0°.6) - 1.77 sin (2 Q+g-36°16') - 0.58 sin (4Q+2g+84°) - 0.50 sin (4 Q + g - 47°) - 0.43 sin (3 Q + g - 47\7) Aberrations - Correction in longitude due to aberration of light in earth's atmosphere is - 20".50 - 0".34 cos g Nutation Tropical longitude is calculated from mean e quinox of the date. Correction due to nutation is *° be made in tropical longitude, but, not necessary w nirayana longitudes. I 248 Siddhdnta Darpan Solar nutation = - I". 27 sin 2L+0M3 sin g - 0.05 sin (3L+79 ) Lunar nutation = - 17". 23 sin Q + 0".21 sm 2Q i Principal term of the lunar nutation is slowly) increasing at the rate of 0".17 per thousand years3 In the obliquity . of ecliptic, Solar nutation = + 0."55 cos2L + 0.02 cos 1 (3L+79°) Lunar nutation = + 9".21 cosQ - 0".09 cos 2Qj Here L and Q are the tropical mean longitud of the Sun and the lunar node respectively. Radius Vector Radius vector is expressed in terms of meanl distance of earth from sun. Mean distance isj expressed by Gauss formula based on Keplar's thircji law aV = k 2 (1+m) -where k is Gaussian gravitational constant * 3548". 187607 m = mass of earth and moon, taking su mass as unity n = observed sidereal mean daily motion o earth. a = mean distance from sun to mass centr of earth and moon. Value of k is based on sidereal period 365.256898 days of earth considered as partid True Planets 14$ without mass or of 365.256344 days with adopted value of mass. With Newcomb's value of m = 1 + 329390 and n = 3548 ,, .19282 / we get log a = 0.000,000,013. Long term effect of attraction of inner planets is equivalent to an increase in mass of sun, to balance it, radius vector a increases. Observed daily motion n remains constant. Elliptic term of radius vector is (equation 11) R = a n + y -. ( 3,3) cosg-&-\ e * )coslg 3 3 1 , - ge cos 3g - -e* cos 4g ] J. 2 q and log R = log* + log (1 + - ) - M [ (L _ 1^ + (e - |e 3 ) cos g + (|e 2 - ge 4 ) cos 2g _l 17 3 . 71- 4 + ^je cos3g + — e 4 cos4g] where M is the modulus of common logarithm s 0.434294. & Taking value of e for 1900, nn^ R = L000 ' 140 ' 5 " °- 016 ' 7 ®> 2 cos e - 0.000,140,3 cos 2 g 8 nn ' 0.000,001, 8 cos 3 g - 0.000,000,7 T + 0000,04,18 (T+0.003T 2 ) cos g + 0.000,000, 71 Cos 8 n nn„ L ° 8 R = °- 000 ' 03 °>6 - 0.007,274, 1 cos g - UU0C 1,091, 4 cos g - 0.000,0015 cos 3 g - 0.000,000,15 i, + 0.000,018,14 (T + 0.003T 2 ) cos g + 0.000,000,46 1 cos 2 g. • ISO Siddhanta Darparut Terms free of T are value for 1900 A.D. Terms with T are secular variation. Effect of planets on radius vector - Due to venus = - 1".12 cos Q + 3".25 cos (2Q+7') + 0".50 cos (3 Q - 1°5) + 0'M8 cos (4Q-2\4) + 0".08 cos (5Q-3 ) 1 Log = - 2".36 cos Q + 6.84 cos (2Q+7') + 1.0S cos (3Q-1°5) + 0".38 cos (4Q-2\4) + 0.16 cos (5 Q-3') Jupiter = + 3".36 cos (Q-l°6') - 1 ,# .91 cos (2Q--13') - -0.13 cos (3 Q + 4°3r) Log = 7.07 cos (Q-l°6') - 4.03 cos (2Q-130-0.28 cos (3Q+4°2r) Due to Moon = + 6". 35 cos D or log = + 13.36 cos D. Sun's semi diameter and horizontal parallax - At unit distance, apparent semi diameter ol sun = 961 ".18 and horizontal parallax = 8". 794. Atl any distance, Semi - diamter = 961M8 /R ! 16'1 / M8+16'M0 cos g+0".27 cos 2 g Parallax = 8 ,, 794/R = 8".79+0".15 cos g. For calculation of eclipse, allowance of 1".5 is made for irradiation. Then true semi diamet at unit distance is 959". 63 Reduction of Rt ascension and declination A = tropical longitude of the sun, a = righj ascension, 152 True Planets d = declination, e = true obliquity of ecliptic t0 equator, , d<3 Sin d = Sin A sin a and — = sin a tan a = tan A. cos e 1 * or a = A- (tan 2 § sin2A- - tan 4 2 sin 4 A + 1/3 tan 6 I sin 6 A ) ^ = - i sin 2 a tan e = - 0.2168 sin 2 a de 2 Sidereal Time - Sidereal time at any instant is defined to be west hour angle of the First point of Aries (Vernal equinoctical point) from the upper meridian of the place. Sidereal time at mean noon (i.e. 12h local mean time) on any day is the right ascension of the fictitions mean sun, which is defined to be the tropical mean sun at moment as affected by mean aberration. At mean midnight, sidereal time is 12h (i.e. 180°) + R.A. of fictitions mean sun for the moment. Sidereal time at Greenwich mean midnight = 6h6 m 47 s ,558 + 8640184 s .542 T+0 5 .0929T 2 where T is Julian centuries of 36525 days from 1900 AD, Jan 0, h or EX Motion in a century = 100 d + h 3 m4 s .542 Motion in a day = 3 m 56 s .5553605 Equation of Time = Local apparent time - Local mean time * 152 Siddhanta Darpana Local apparent noon = 12 h L.M.T - equation of time (E) At h E.T., Equation of time = Apparent sidereal time - Apparent R.A. of sun. E = R.A. of mean sun - R.A. of true sun Both are affected by aberration and nutation. True sun is also affected by perturbation. Omitting aberration and nutation from both sides, only perturbation A remains in true sun. True Sun = L + equation of centre E = L - (L+Eqn of c) + tan 2 ~ s * n 2A 1 e i E - — tan 4 ^sin 4 A+ — tan 6 '-* sin 6 A - effect of perturbation in longitude Equation of centre in seconds of time is + (460.67 - 1.149T) sin g + 4.82 sin 2g+0.07 sin 3g Value in arc is, tan 2 | = 0.0430836-0.0000491 In seconds of time, tan 2 | = 592.44r-0.675 T So equation of time (in seconds of time) is = - (460.67-1.149 T) sin g - 4.82 sin 2g - 0.07 sin 3g + (592.44 - 0.675 T) sin 2A - 12.76 sin 4A + 0.36 sin 6 A - — perturbation in longitude. 15 m A i ■-■.'I ■$ True Planets 153 4, Equation for other planets Basic constants of Mercury Mean longitude, L for 3200 BC, Jan 0.5 epoch is L = 49°. 677936 + 538106654".8 T-1".084T 2 L for 1900 AD epoch is 173°. 303523 (51 centuries - 13 days) Mean anomaly, g for 3200 BC is g = 53°. 107661 + 538101055.04T - 0".024T 2 g for 1900 AD is 98°. 169610 Argument of latitude, U is for 3200 BC U = 62'. 977228 + 538102388".05T-0".458 T 2 U for 1900 AD is 136 '.609863 Constants for venus - Mean longitude L for 3200 BC (-51 centuries + 13 days) = 285\18561+210669162".88T+1".1148T 2 L for 1900 AD is 341°. 97032 Mean anomaly g for 3200 BC is g = 223°. 83111+210664093.95 T + 4".63 T 2 g for 1900 AD = 214°. 34622 Argument of latitude U for 3200 B.C. is U = 252°.31206+210665923".42T+0".3612T 2 U for 1900 AD is 266.59425 Constants for Mars - for 3200 BC L=33°.370172+68910117".19 T - 1".1184T 2 „ g = 152° 99708 + 68903493. 19T-0".651T 2 - 0'\0192T 3 154 Siddhdnta Darpana U = 23°. 923117 + 68907340."7 T - 1M234T 2 - 00".00192T 3 For 1900 AD epoch constants are L = 292° 416147, g = 318°. 387964 U =242°. 918470° Constants for Jupiter - For 1900 AD are L = 238°.0496+10930687".148T+l".20486T 2 - 0".005936T 3 U = 138°.60587+10927049".24T + 0".06314T 2 + 0".024704T 3 g = 225°.32833+10924891".286T+2".59772T 2 + 0".06314T 3 Long period inequality in longitude L is E = (1186". 618572 - 0". 0347004 t + 0".000033372t 2 ) SinC - 12".013596 sin 2C where t is number of years from 1800 A.D. C= 95°. 8814+0°. 38633184 t + 0°. 000035 It 2 Constants for Saturn - for 320Q BC are L = 147°.9623+4404635".581T-1".16835T 2 -0".021T 3 g = 156°.74269+4397585".284T-1".80655T 2 - 0"0376T 3 U = 79°. 704558+4401492". 0785T-1".7162T 2 - 0".0019T 3 Constants for 1900 AD are L = 260°. 46036, g = 172°. 74219, U = 152°.43062 Notes : (1) Newcomb's formula has been corrected by Rossi for Mars. There are other perturbations for planets also which have not been written. True Planets 155 (2) Moon's motion in detail will be discussed in the next chapter. (3) From these constants, equations of centre and radius vector can be obtained. (4) Value of eccentricity also changes. For the present century, values given in chart can be taken as constants. (5) From these equations, constants are tabu- lated for centuries. Then by ratio, they are fixed for specific years. (6) Equations of centre and radius vector give true positions from the constants of year. (7) Longitudes and latitudes are reduced to ediptic. (8) Heliocentric longitude and latitude are converted to geocentric values. Let S he longitude of Sun, R its radius vector H is heliocentric longitude of planet, b its latitude r is radius vector from sun of planet, x is geocentric longitude, y is latitude r cos b sin (H - S ) _ then tan P = — . V m — "T , x = S + P R + r cos b cos (H - s) r sin b sin P tany = R + rcosbsin(H-S) 5. References - (1) Any text book on modern coordinate geometry, Trigonometry can be referred for these formula. (2) Dynamics of planetary motion has been explained in Dynamics of Rigid bodies by A.G. Webster. 156 Siddhanta Darpana (3) Derivation of formulas can be referred in books on spherical trigonometry. Inportant books are (1) Spherical Trigonometry by Gorakh Prasad, Pothishala, Allahabad - 2, (2) A hand book of Practical Astronomy by R.V. Vaidya, Payal Prakashan, Nagpur (3) Astronomy by G.V. Ramachandran, Tiruchirpalli (4) Practical Astronomy by Schroeder, published by Werner Lauries, London (5) Astronomy by R.H. Baker - D van Nostrand, East West Edition (6) Celestial Dynamics . by W. Smart - Longman, Green (4) Astronomical charts were published by Simon Newcomb in 1899 and 1906 but they are out of print. Nautical Almancs published by govt, of India, specially 1st edition of 1958 can be referred. Tables of sun have been published by Sri N.C. Lahari from Calcutta in 1993 (revised edition.) Translation of the text Verse 1 - Scope and definition - The position in which graha in seen from earth is to be found by calculation. This process is called Sphutikarana of graha or making is sphuta. The graha give results according to the position they are seen. Hence method for making a graha sphuta is being explained. Verses* 2-5 - Reasons of planetary motion - Celestial sphere containing graha and naksatra revolves around earth once in a day from east to west due to attraction of a wind (Pavana) named Pravaha rotating round earth. This is called daily motion. Graha move in opposite direction from west to east compared to stars (or naksatras) with slow speed according to their own enregy. This is called natural speed of a planet (svabhavika gati). There True Planets ^7 are deviations from this average speed of planets under influence of ucca (sighra and manda). (According to Surya siddhanta) Invisible forms of kala like sighrocca, mandocca and pata residing in celestial sphere are reasons of planetary motion. Notes (1) Daily moiton is due to rotation of earth around its axis. Earth appears fixed to us and stars rotate in opposite relative motion. Reason of earth's rotation is due to initial conditions of its formation, now it continues due to inertia. That inertia is assumed to be 'Pravaha', an imaginary force in vacuum. This is similar to assumption of ether for propogation of light in vacuum. (2) True position of a planet is closer to mandocca (farthest point in elliptical orbit) com- pared to its mean position. Due to that reason attraction by mondocca in seen. Similar is case with sighrocca. (3) Pata is point of intersection of planetary orbit with ecliptic due to its inclination. Hence, pata appears to repulse a graha away from ecliptic. Verses 6-9 - Nature of motion - Mean sun moves around earth between naksatra and orbit of grahas. Other planets like mars are in orbit round mean sun and along with it, they also revolve round earth. Hence (mean) sun is called attractor of all. From dainika gati of Mafigala, Brhaspati and sani - dainika gati of ravi is more and they are attracted by ravi. Hence ravi is called sighrocca of these planets. Compared to Budha and sukra, speed of ravi is slower and it always remains between them. Hence Budha and Sukra are called their sighrocca. 158 Siddhanta Darpana Notes (1) Outer planets are almost in same direction from earth as from sun. Minor correction is due to position of sun from earth. (2) Inner planets are within a small distance from sun which is their average position. First correction for their true position, is due to their own motion. Hence they are own sighrocca. Verses 10-16 - Slow, fast and reverse motion - Planets in successively farther orbits from sun are - budha, sukra, mangala, brhaspati and sani. Hence their angular speed appears progressively slower from earth (if linear speed in orbit is assumed to be same). Like ravi, moon also is rotating round the earth, but from very close distance. Hence angular speed of moon is largest, though its linear speed (in yojanas etc) is small. Budha and sukra are close to ravi, compared to earth. Therefore, they are seen with ravi after 12 rasi (full rotation), as well as, after 6 rasi (half rotation). Mangala, brhaspati and sani are farther from ravi - compared to earth. Hence, they appear together with ravi at 12 rasi * difference and in opposite direection at 6 rasi difference. When earth is in one direction of ravi, and star planets (tara graha) mangala, budha, guru, sukra and sani are in opposite direction - then the graha appears to move in forward direction (margi gati) If a tara graha and earth are in the same direction of ravi, then the graha appears to move in reverse direction (vakri gati) due to difference between mean and sighra speeds. (Figures 7, 8) 159 True Planets Notes (1) Explanation of forward and reverse speeds - Figure 7 indicates relative speeds of earth 0* Vernal equinox 180' Figure 7 - Foreward and reverse speeds of inner planet (Budha) ■ * Vernal equinox 270" 180' Figure B - Foreward and reverse speeds of outer planet (Mars) and an inner planet Budha. Figure 8 compares earth's motion with an outer planet mars. Numbers 1, 2 8 indicate successive simultaneous positions °f the planets in their orbits. Ecliptic is a much bigger circle whose 0° stars' from vernal equinox. AU the circles are in same plane and the movements ar e in positive direction (anticlock wise). Line of :m ■y 260 Siddhanta Darpana sight from position 1 of earth to simultaneous position of planet 1 on a point on ecliptic is marked 1. Point 1, 2 — 8 on ecliptic are apparent directions of the planets seen in ecliptic after regular intervals. Speeds in inner orbits are faster. Figure 7 shows that 1,2,3 positions indicate forward movement of Budha. After point 3, budha comes on same side of sun as earth and moves backwards in positions 4 and 5. Between 5 and 6, it is almost stationary before moving forward again to positions 7 and 8. Similarly, positions 1 to 5 in figure 8 indicate forward motion of mars. At 6, it moves back wards when earth and mars both are on same side of I sun. Actually ecliptic circle is at almost infinite distance and position 7 also is in backward direction from 6. But due to small construction of ecliptic it looks forward. From position 8, planet again moves forward. (2) Fast and slow speeds - Angular speed is arc length divided by radius in unit time. Hence for same arc length in unit time, angular speed will be less for large radius. Thus farther planets will look slower. In siddhanta text, this was considered only reason for slow speed. But linear speed also becomes slower as explained by Keplar's law (to balance lesser gravitational pull). Verses 17-18 - Five tara graha revole round the sun at constant distance in east direction. They are attracted by their mandocca and sighrocca ravi. They always* remain in bha-cakra (ecliptic circle). Earth is in centre of bha-cakra, ravi is at the centre of five tara graha. Hence at full circle (12 I .•j True Planets 161 rasi) or half circle difference, earth is in same line a s sun and the planet. Verse 19-22 : At 3 rasis after cakra or cakrardha (90° after 360° or 180° - i.e. 90° or 270°) i.e. at the end of odd quadrant, difference between planet's direction and sun's direction is maximum. Hence sighra paridhi is different at the end of odd and even quadrants (0°, 360"). Circle of naksatras (bhakaksa) is 360 times away from its centre earth, compared to distance of sun's orbit from earth. Division by this ratio (hara) 360 into degrees of 1 revolution (360°) we get 1° which is difference between sighra paridhi at the end of odd and even quadrants. I (author) wil explain the difference between mca and ucca paridhi. This can be seen direectly by observation, so presumption is not necessary. Explanations (1) Difference in sighra paridhi at end of odd and even quadrants is due to elliptical shape of planetary orbits. The difference depends on eccentricity of the orbit. It will be explained later on while explaining motion on sighra and manda paridhis. (2) The assumption is that difference in points of observation causes difference in paridhis. At 90° or 270° sighra kendra, difference is maximum due to sighra motion. Difference in heliocentric and geocentric position will be angle made by radius of sun orbit at distance of star circle. Difference of 1° will be observed from 57.3 times the distance. This assumption will be correct if stars orbit is considered 60 times away compared to earth's orbit. 162 Siddhdnta Darpana This figure has been accepted by Aryabhata, Surya siddhanta and all others. 360 times the distance will give less than 176 difference. Reasoning is wrong in both ways - about reason of difference in sighraparidhi and about the angle of difference. (3) Siddhanta darpana has taken sun's dis- tance about 11 times the figure accepted in classical siddhantas (based on 72,000 yojana diameter in Atharva veda is stead of 6,500 yojana in siddhanta). He has increased distance of stars further 6 times. Even after increase of about 66 times, it is still highly under estimated. Even the nearest star (4.4 light years) is 2,80,000 times the distance of sun. Rohini at 14 lakh times and svati about 87 lakh times the distance are other nearest stars. Verses 23-27 * Attraction of ucca - Planets starting from ravi (all) are attracted by gods named their mandocca by chord of air (invisible force of attraction to an imaginary point mandocca). Hence true planets (spasta graha) are always deviated towards mandocca from their mean position (mad- hyama graha). Planets with slow motion of own and attracted by their mandocca, also move under influence of pravaha from east to west (due to daily motion of earth). (From Surya siddhanta) - Planets being always (day and night) under attraction of their ucca, move in different ways - sometimes east or west. When ucca is in east semidrcle of the planet, ucca pulls , the planet towards east. When ucca is in western -j semi circle, it pulls towards west. Direction of ucca on a circle is always same, but it is called east in True Planets 163 one semi circle and west in the other. When the planets move towards east under attraction of ucca, the deviation is positive and in west it is negative. (Normal motion of planets relative to stars is towards east hence deviation in same east direction is added and in opposite direction, substracted.). Explanation - (1) Actual orbits of ravi and candra are elliptical round the earth (relative motion of ravi). In such an orbit earth is not at centre but on a focus. Thus centre of planetary motion is deviated towards farther end of major axis called mandocca. It is called so because at this position candra is farthest and hence slowest (ucca and manda). To a first approximation mean planet moves in a circle round earth at the focus. Next approximation assumes motion in an eccentric circle with centre at centre of the elliptic orbit.All the points of this circle are thus deviated from corresponding position of madhya graha towards mandocca. Hence mandocca appears to attract the planet. More accurately, mandocca doesn't attract because in such case, speed of the planet will be increasing in that direction. It should be maximum at mandocca. But it is only a deviation or displacement towards mandocca. This will be explained mathematically while computing the corrections. (2) If we rotate along a circle/ after reaching mandocca, the movement will be away from it upto 180° difference. In remaining half it will be towards mandocca. Since attraction is always towards niandocca, 0° to 180° circle after it is considered negative or western deviation. 164 Siddhanta Darpana ■■< Verses 28-33 - Viksepa and pata - Orbit of j planets starting from moon (except ravi) are at an j angle with the kranti vrtta (i.e. apparent orbit of j ravi). It meets kranti vrtta at the points, where the \ circles bisect each other (being great circles of a '; sphere). Half of the orbit is north of kranti vrtta (upward direction of right hand screw rotating in j orbit direction). Other half is south. At pata, the planets are on kranti vrtta; Away from pata they are deflected towards north or south slight from Kranti vrtta. Hence pata is considered reason of north or south deflection called viksepa (or sara). Sara is distance of perpendicular from graha on Kranti vrtta, measured in kala or minutes of angle). Foot of perpendicular is manda sphuta graha. (From Surya siddhanta) - When planet is ahead of its northern pata by 0° to 180 °, it is deflected northwards. (Hence this pata is called northern pata or pata, in short). Then pata is behind the graha and called in west. Pata in east (or before the graha) deflects it towards south. Budha and sukra revolve with sun (being in inner orbit). Their sighrocca is the planet itself. So pata east from sighrocca causes south sara. In other half of orbit it is north sara. When pata is with graha or 180° away, graha is on one of the pata points and hence on the kranti vrtta. There is no sara in that position. When difference between planet and pata is 90° or 270 (end of odd quadrants), sara is maximum (parama). Mean parama sara of planets is given below (compard with modern values given in introduc- tion) True Planets ibl planet Sara in Kala Degree Modern Value Candra 309 S'9 7 5°8'42" Mangala 111 1*51' 1*51'0" Budha 164 2°44' 7*0*14" Brhaspati 78 1*18' 1*18'21" Sukra 148 2*28' 3*23'39" Sani 149 2 C 29 / 2°29 / 25" Notes (1) Siddhanta darpana figures are an imprpvement over previous siddhanta books ac- cording to comparative chart given below. Except two inner planets, this compares well with modern values. (2) Pata according to other texts ■ Surya Brahma sphuta Mafia Ptolemy Siddhanta Siddhanta & siddhanta Siddhanta Siromani • Candra 4°3(T 4°30' 4*30' 5*0' Mangala 1"30' 1*50' 1*46' l'C Budha 2*0' 2*32' 2° 18' 7*0' Guru VV 1*16' 1*14' 1*30' Sukra 2*0' 2*16' 2*10' 3*30' Sani 2°cr 2*10' 2*10' 2*30' (3) While figures of siddhanta darpana for moon and outer planets are very accurate, it looks wrong for budha and sukra compared to modern figures. Reason is that the modern figures are heliocentric whereas these are geocentric. In this figure 9, E is earth centre, S is sun centre and ESA is Kranti Vrtta, cut by plane perpendicular to budha orbit. This plane cuts 166 Siddhanta Darpana B' B Figure 9 - Pata of inner planet budha orbit at points B and B'. As seen from sun, budha orbit makes angle BSE or B'SA with kranti vrtta. From sun this is parama sara. As seen from earth, parama sara is ZB'ES or Z.BES (almost equal because budha orbit is small). If ES is taken as 1, then mean radius of budha orbit SB = SB' = 0.3871. sin L B E S _ sin L B S E BS " BE B S or sin L B E S = =-= x sin L B S E d h 0.3871 « = — - — x sin 7° o' 14" = .3871 x .1219= 0.0 472 or L B E S = V 42' which is very close to value for budha in this book (2° 44') Relative distance of Sukra is 0.7233 hence its angle is given by (as seen from earth) Sin 6 = .7233 x sin 3°23'37" = .7233 x .0592 = .0428 or = 2° 27' which is only V less than siddhanta darpana. Verses 34-41 - Types of planetary motion - Motion of planets as seen from earth's surface is called sphutagati. Sphuta gati is of three types - True Planets 167 forward (prak or margi)) gati, reverse (vakri) gati an d zero motion (sunya gati). Forward motion is of five types, reverse motion of two types and one zero speed - these are eight types of motions of planets starting from mangala (Ravi and candra have no reverse motion). From Surya siddhanta - Eight gatis are named . 1. Vakra 2. Anuvakra, 3. Vikala, 4. manda, 5. mandatara 6. sama, 7. slghra and 8. Atisighra. Reverse motion starts reducing in later half, it is called vakra. At the start of reverse motion it is increasing. Then it is called anuvakra. When forward motion is less than mean speed and is still decreasing, it is called mandatara,. when it is increasing, it is called manda. Spasta gati equal to mean speed is called sama. Spasta gati more than mean speed and still increasing is called slghratara. When decreasing it is called slghra. Ravi and candra are affected only by mandoc- ca (not slghrocca). They have only five types of speeds - 1. manda, 2. mandatara, 3. sama, 4. slghra and 5. sighratara. Their meanings are explained above Verse 42 - Sphuta method - (From surya siddhanta) Now, I tell will respect, the method of making a graha sphuta by calculation, where mean planets arrive due to 8 types of speeds at the observed place, (drk-tulyata = calculation equal to observation). Verses 43-46 - Explanation of arc and its sine - Now, I tell the method of calculating skue and 168 Siddhdnta Darpana arc, which is used in many sciences and by knowing which, people get the title of acarya (doctorate). As a cloth is interspersed with threads, a gola (sphere or its circles) is also mixed up with sines and versed sines, (sine can be found between any two points of a sphere or circle and hence they are infinite in number). To find the sine (jya) of a radius inclined with starting radius at 0°, we make a jya of same arc in opposite direction. Graha is on top of jyarddha (end of radius vector) with which calculation is made. It is also called jya in short. Half part of a circle or full revolution (bhacakra) looks like a bow (capa). The bisecting line of circle passes through its centre and is called diameter (vyasa). Verses 47-54 - Method of Calculating sines - For 3 raSi (90°)jya, Kotijya have extreme values. Jya of 3 rasi passes through centre and is equal to radius. Then it has greatest value, Kotijya is distance of this jya from centre and is here (for 90°). 96th part of a circumference is very small and almost a straight line. Hence it is almost equal to its jya. Thus 1/8 part of a rasi lipta (1800) i.e. 1/96 of circle is 225 and is equal to first jya. First jya is first khandantara (difference) (i.e. Jya of 225' - jya of ; )i To find the second Khandantara (2nd jya - 1st jya), 1st jya is divided by itself and result (1) is deducted from 1st jya 225 (225 - 225 = 224). Result is 2nd khandantara. Add 2nd khandantara in 1st jya to get the 2nd jya (i.e. sine of 2x225' arc). 2nd jya is again True Planets 1*9 divided by 1st khandantara. If remainder is more than half of 225 then it is ommitted. Quotient deducted from 2nd khandantara gives 3rd Idiandantara. 3rd Khandantara addeed to 2nd jya, we get the 3rd jya. Similarly jya pinda (quantity) is divided by 1st jya and substracted from its khandantara to give next khandantara. This way we get the jya of 1st to 24th jya pinda in lipta. In dividing 6,7,12,15,17,20,21 jya pinda, reemainder is more than half of divider 225. But we still omit it without adding 1 to quotient because Brahma had told so to Narada. Notes : (1) Narada Purana also gives a complete summary of astronomy and astrology. In the chapter describing calculation of sines, no such explanation from Brahma has been given as stated here. However, the stated values have been given which means the same thing. There is no dialogue from Brahma in the chapter, but he is considered original source of the knowledge. (2) Increase in sines is proportional to its d differential coefficient thus — (sin x) = cos x is ax proportional to 1st difference (khandantara) Change in difference itself; i.e. 2nd diff is proportional to 2nd differential coefficient. Thus 2nd difference = d 2 d (cos x) . . . "Tj (sinx) = — H = - sin x. (proportional) Let 1 part be P = 225'. Hence sin x = Sin nP, n=no. of parts. Here jya = R sin x where R is radius equal to 3438 kala. w First difference = Ai (or 1st khandantafa). 170 Siddhanta Darpana d Ai= .— (R sin x) . Sin P = R (Cos x) Sin P = R sinP. c j jvcf ■ r* & R<5x.sinx Second difference S = -7- (R cos x) = - dx x ' R Negative sign means that the first differences (khandantaras) are decreasing with increasing angle and 2nd differences are proportional to jya (R sin x). It is to be divided by R to get modern sine. (R cos x . d x) _ -R sins x . dx 2 R - tf a* onO -. - , d ** 225X 225 At x = 90 it is equal — — = — rrrr M R 3438 = 14 / 43"30 / ". This has been explained by Sri Ranganatha in his tika on Surya siddhanta called Gudhartha - Prakasika. But he has taken this as 3438/225 = \5'\6"^'" by mistake. Even this is approximate and correct value is 14'47" (3) Sri Bapudeva Sastri has given the following proof of the formula in his English translation of Surya siddhanta Ai = Sin P - Sin 0° A 2 == Sin 2 P - Sin P " A 3 = Sin 3 P - Sin 2 P An = - Sin n P - Sin (n-1) P An+1 = Sin (n+1) P - Sin n P Then Ai - A 2 = 2 sin P - sin 2 P = = 2 sinP - 2 sin P cos P = 2Sin P (1 - cos P) = 2 Sin P. ver sin P (versed sine = lrcosine = utkrama jya.) A 2 ~ A 3 = 2 Sin 2 P - Sin P - Sin 3 P True Planets I 71 = 2 sin 2 P - sin P - (3 Sin P - 4 Sin 3 P) = 2 sin 2 P - 4 Sin P + 4 Sin 3 P = 2 sin 2 P - 4 Sin P (1 - Sin 2 P) = 2 Sin 2 P-4 Sin P. Cos 2 P = 2 Sin 2 P - (2 Sin P Cos P) 2 cos P = 2 Sin 2 P (1 - cos P) = 2 sin 2 P. versin P A 3 - A4 = 2 sin 3 P - Sin 2 P - Sin 4 P = 2 Sin 3 P - 2 Sin 3 P. Cos P = 2 Sin 3 P (1 - Cos P) = 2 Sin 3 P versinP An- A n + i = 2 Sin n P-Sin (n-1) P - Sin (n+l)P = 2 Sin n P - 2 Sin n P Cos P = 2 Sin n P (1 - Cos P) = 2 Sin n P versinP. Adding the above equations, we get Ai - A n + 1 = 2 versin P (Sin P + Sin 2 P + Sin 3 P + — Sin n P) But Ai - A n + 1 = Sin P + Sin n P - Sin (n+1) P Hence, Sin P + Sin,<nP - Sin (n+1) P = = 2 ver sin P (SinP + Sin2P + + Sin nP) or Sin (n+1) P = Sin nP + Sin P-2 versin P (Sin P + Sin 2 P + — Sin nP)) Here P = 3 a 45' = 225' *. 2 versin P = 2 ver sin 225' = 2 (1-Cos 225') 2(1 -0.9978) = 2X0.0022 = ^ = ^ = ^ approx. Thus sin (n+1) P = Sin nP + Sin P - 1/225 x (Sin P + Sin2P+ -- Sin nP) 172 Siddhdnta Darpana This is the formula for finding (n+l)th sin from nth sin i..e sin np, First khandatara is added and sum of pervious sines divided by (225) is substracted. (4) Bhaskara II has explained that the sines were found by constructing regular polygons of increasing number of sides in a circle. Aryabhata has indicated the geometrical method for finding sines for 12 divisions of a right angle (7°30 / each) in a circle of radius R = 3438'. (Method is explained by Prof. Kripa Sankara Sukla). Let figure 10 represent a circle of radius R = 3438'. Divide the quadrant into two at T (45°) each. Trisect TA into TB, BR, RA (15° each), RA into L Figure 10 - Geometrical method of sine table two (RQ, QA, 7-1/2° each). Mark off AL = 30°. Join LB. This is equal to R and denotes chord 60". 2^ R sin 30" = — = m^ 1° This is the 4th sine, in the 7 ■=- table to" be 2 computed New, from right angled A OMB, True Planets 173 OM = V R 2 _ (H / 2 )2 = v1>R = 2978 This is R sin 60°, i.e. the eighth R sine Now from rt angle AAMB AB = V (R sin 3q0 f + (R vers 3q0 f = V(i7i9)2 + (460) 2 = 1780 This is chord 30°. Half of this i.e. AN, is R sin 15° Thus R sin 15° = 890' This is second R sine Now from rt angle AANO ON = V A0 2 - AN 2 ) = V R 2 _ (R sin ^ = 3321 This is R sin 75° = the tenth R sine Now in rt AANR,, where R is mid-point of arc AB, we have AR = V AN 2 + NR 2 = ^(R sin 15' ) 2 + (r versl5° ) 2 ^890 2 + 117 2 - = 898' This is chord 15°. Half of this i.e. AS is R sin 7°30' Thus, R sin 7°30' = 449 / This is the first R sine Now, in rt A ASO, OS = V R 2_ (Rsin ,70 30) 2 = 3409' This is 11th R sine for angle 82°30 / Now, R ver s 75° = R-R sin 15°, s o that chord 75" = V( R sin7 5 ) 2 + (Rvers75° ) 2 = 4186 ' Half of this 2093' is R sin 37° 30' (fifth R sine). 174 Siddhanta Darpana Now, R sin 52-3C = VP~T^~~"^1^=2728 (7th R sine) In semi square AOD, OA = OD=R so AD = V2 R = 4862' This is chord 90°. Half of this 2431° (i.e. AP) = R sin 45° In A AFT, AT = V (R sin 45 <> f + (r vers ^2 , 2630- This is chord 45°. Half of this is R sin 22*30' = 1315' (3rd R sine)) R sin 67-30' = V R 2 _ (R sin 22 ° 30) 2 = 3177' (ninth R sine). These are 12 R sines. By finding chord of 7° 30' arc we can find R sines of 3° 45' intervals also. (5) More accurate method is to calculate sine by infinite convergent series. Sin 0= 6 - 2- + ^- + Z3 Z.5 where 6 is expresseed in radians (arc/radius) and is betweeen 0° and 90°. verses 55-66 — All the verses are quoted from Surya siddhanta. Verses 55 - 60 - These tell the values of 24 R sines at intervals of 3° 45' in kalas. Next verses give values of utkrama jya = R (1-cos 0) No. Arc R Modern Diffe Vers Diff. Modern valul sines values rences radius=l 1 1 1 225' 225' 224.856 225 7 7 .0022 ] 2 450' 449' 448.749 224 29 22 .0086 \ 3 675' 671' 670.720 222 66 37 .0192 4 900' 890' 889.820 219 117 51 .0341 I 5 1125' 1105' 1105.109 215 182 65 True Planets 175 6 1350' 1315* 1315.666 210 261 79 .0761 7 1575' 1520' 1520.589 205 354 93 .1031 8 1800' 1719' 1719.000 199 460 106 .1340 9 2025' 1910' 1910.050 191 579 119 .1685 10 2250' 2093' 2092.922 183 710 131 .2066 11 2475' 2267' 2266.831 174 853 143 .2481 12 2700' 2431' 2431.033 164 1007 154 .2929 13 2925' 2585' x 2584.825 154 1171 164 .3406 14 3150' 2728' 2727.549 143 1345 174 .3912 15 3375* 2859' 2858.592 131 1528 183 .4445 16 3600' 2978' 2977.395 119 1719 191 .5000 17 * 3825' 3084' 3083.448 106 1918 199 .5577 18 4050' 3177' 3176.298 93 2123 205 .6173 19 4275' 3256' 3255.546 79 2333 210 .6786 20 4500' 3321' 3320.853 65 2548 215 .7412 21 4725' 3372' 3371.940 51 2767 219 .8049 22 4950' 3409' 3408.588 37 2989 222 .8695 23 5175' 3431' 3430.639 22 3213 224 .9346 24 5400' 3438 3438.000 7 3438 225 1.0000 Notes (1) Difference for versed sines are in opposite order and they need not be calculated. From them versed sines are calculated. (2) Modern values of sin, cos and other ratios are calculated for radius 1. Hence, for calculating Indian sines they are to be multiplied by radius. (3) Madhava method for calculation upto 9 decimal places - This has been quoted by Nflakantha in his commentary on Aryabhatfya. His sentences indicating calculation parameters have been quoted by Sankara in his commentary on Tantra sangraha by NHakantha. Original book of n p .*:.< Siddhdnta Darpana Madhava is not available. He must have useJ infinite series and then formed the simplified ruleJ expressed by verses in 'Katapayadi' form. ^? Method for sines - Place the expression*! 0'0"44"', 0'33"6"', 16'5"41"', 273'57"47"', anl 2220'39"40"' - five numbers from below upwards^ Multiply the lowest by the square of the choseij arc and divide by R 2 (i.e. 2,91,60,000 = 5400 2 )! Substract the quotient from expression just above}, Continue this operation through all the expression! above. The remainder got at last operation is m be multiplied by the cube of the chosen arc an divided by R 3 (i.e., 157,46,40,00,000). Substract t quotient from the chosen arc to get its R sine. Method for versed sines - Place the si expressions - , /, 6' // ,0 / 5'12 /, / 3'9"37"' 71'43"24 ,i 872'3"5'" and 4241'9"0"' from below' upward Multiply the lowest by the square of the chose arc and divide by R 2 . Continue the operatio; through all the opeations above. The last quotie* will be the versed sine of the chosen arc. Thi formula is based on series for sin upto term n Results No. Arc. R sine sine in decimal 1 225 224'50"22"' .06540 2 450 448 , 42"58 , " .13053 3 675 670 , 40"ll m .19509 4 900 889'45"15"' .25882 5 1125 1105T'39"' .32144 6 1350 1315 , 34 ,, 7"' .38268 7 1575 1520'28 , '35" .44229 Modei Value .06540 .13053 .19509 .25882 .32144 .38268 .44228 True Planets 177 8 1800 9 2025 10 2250 11 2475 12 2700 13 2925 14 3150 15 3375 16 3600 17 3835 18 4050 19 4215 20 4500 21 4725 22 4950' 23 5175' 23 5400* .50000 .50000 .55557 .55558 1718 , 52"24 , " 1909 , 54"35"' "2092'46 w 3 m 2266'39"50 m 2430'51 ,, 15 m 2584 , 38 M 6 m 2727*20"52 m 2858'22"55 m 2977'10"34 m 3083'13 w 17 m 3176'3"50 ,n 3235 , 18"22" 3320'36"30" ? 3371 , 41"29 m 3408 , 20 w ll m 3430'23"ll m 3437'44»48 m Verses giving value of constants in katapayadi constants for sine - (with method) fa*c*llPcW: WF$ t #w|ijM^q4«T *ftfc s^^d: II Constants and method for versine - '#T:' ^ftfr?p:' ^^^Tf^gT ^T^T^^; 'TOf ^w4t ' 3TT«R^T^TjfnRfT <?% «Fpi: ^H fa^lfcW: Sine table in paras - ,60876 .60876 .65934 .65934 .70711 .70711 .75184 .75184 .79335 .79335 .83147 .83146 .86603 .86603 .89687 .89688 .92388 .92388 .94693 .94692 .96593 .96593 .98079 .98079 .99144 .99144 .99785 .99785 1.000001.00000 178 Siddhdnta Darpat '^^^^''qm^^«^' im 11 '^^^■^^^"Hftwereft^'.ll^ll cicH<lR^^pm^^TT^r5^Tn«T%f^: |V9 I Proof of the method for sines - Madhava has used the infinite convergent series for sine for 6 expressed in radians (betweei and nil) L3 L5 [7 L9 |n (Terms upto 6 n have been used for desired accuracy) e 3 e 5 e 7 e 9 h 1 or sin# = $ + 6 120 5040 3, 62, 880 B n 1 3 99, 16, 800 " (1 ) Constants of Madhava expressed in para are 'H frue Pl anets 179 (1) 79,94,380 = Ai (2) 9,86,267 = A 2 (3) 57,941 = A3 (4) 1,986 = A4 (5) 44 = A 5 from up to down order. Let x is the arc length in minutes (kala). Then 6 converted to degree (x 180/n; ) and in minutes becomes x 180 10800 orx = 6x — x60 = -^-e At each stage we take its square and divide by R 2 . (R = 5400) i.e. multiply by a 2 where a = x_ R x 10800 n 1 20 _ _ — — x 6 x — — - = — « a (2) R n 5400 » w - After multiplying 5th quantity by a and substracting from 4th, we get A4 - a A5 2 Multiply this by a and substract from A3, we get A3 - a A4 + a A5 2 Multiply by a and substract from A2, we get A2 - a 2 A3 + a A4 - a 6 A5 Multiply this by a and substract from Ai, we get Ai - a 5 A2 + a* A3 - a 6 A 4 + a 8 A5 Multiply by a 3 and substract from arc x - 10800 — jf- we get x - a 3 Ai + a 5 A 2 - a 7 A 3 + a 9 A4 - a 11 A 5 This is value of sin 6 in arc length of minutes. To get it in ratio for radius 1 we have to divide * by 10800/;*. Then Sin 6 = j^- ( X . a 3 Ai + a 5 A 2 - a 7 A3 + a 9 A* - a n A 5 ) - - (3) 180 Siddhanta Darpana First term ■■ tq§qq x x = 6 radians «. ■ . % fa) 3 79,94,380 Second term = ID g gg x \^J ^ (A is divided by 3600 to convert it in minutes) - — approx. (taking n = 3.151926 ) Similarly we get all the terms of series (1) from formula (3) by calculations. (4) Vatesvara has used 96 divisions of a 1 quadrant, each division being 56'15" of arc. He hasj given values of R sine and R versine in seconds! of arc. i Murusvara has taken radius length as 191 and j at 1" intervals, given the values of R sines uptoj 4th division of a degree (upto 1/60x60) of a second). | Kamalakara and Jagannatha Samrata both have! taken radius of length 60 and given values upto j 5th division of a degree. They have taken intervals of 1" and 1/2" respectively. (5) Direct computation of R sines - Madhava formula an be used fbr any angle for calculation upto 9 decimal places. Bhaskara I, Brahmagupta, Vatesvara and Srfpati have given formulas for direct calculation. All of their formula are equivalent to; the following expression. e (180 - 0) Sin = — — — — ; $ in degrees. 10125.- - 0(180-0) While explaining calculation of sine ratios - upapatti vasana), Bhaskara n> has given two foi of formula which Reduce to the same expression. Proof : In Figure 11, C A is diameter of a True Planets circle of radius R Arc AB = 0° and BD = R sin 1 Area ABC = - AB. BC 2S1 Also, Area ABC = - AC. BD So AC BD AB . BC so that AC Let BD (Arc AB) (arc BC) 1 xAC BD ~" f*rr Am r*^nr\ + Y 2xR (arc AB) (arcBC) (180 - 0) zr^ + y or Rsin0 Figure 11 2xR + g(180~-fl).-y 0~(m - 0) or P-^p- ^(180-6) 2xR + 19(180 -0)y (1) 182 Siddhdnta Darpana ■ « a 1« 30 x 150 Putting = 30°,-R = 2xR + 30 x 150 y 9000 or 2R + 4500 y = -z- (2) Putting 6 = 90°, \\ 8100 2xR+ 8100 y = — - - - ■- - (3) 1 ■ „ 4050& Form (2) and (3) y = - — and 2xR = — -=- v ' v ' J 4R 4 R h ■ 0(180-©) ' I Hence from (1) Sin © =. — — "i 10125-7 6 (180 -6) I Alternate proof : a + bA .+ cA 2 I Let sin A = — r ti A + BA + CA 2 I where A is in radians and corresponds to flf degree ','*'* Putting A = 0, a = "■■■" '„ '& Putting A = Jt, b + jt c = So c = -^ bA (n '— A)/tt Thus sin A = A + BA +CA z Since Sin A = sin ( jt - A ), bA (x - A)/jt _ frA (x - X)/x A + BA + cA 2 A + B(^ -A) + C (^ -A) 2 or, A + BA + cA 2 = A + B (x - A) + c (jt - A) 2 or, B (2A - x) = C x (x - 2A) or, C = - - ■%i ■s True Planets 183 Therefore, Sin A = - — .**, 1N A 7i + B A (it - A) Putting A ■» - jr (Sin A '*-). A« + B 7 «(jr-f) a5 '2b.f (»-5) 5;r 2 B 10;r 2 b cr,A» + — -■ = -3P (4) Putting A - I (Sin A = 1) A*: + B 2 (* - 2) = — (jt - j) . 1 2 „ 1 2 or A jz + -7T B = - jT b (5) 4 4 From (4) and (5), B = - 7 b, A = -^ 4 16 Therefore, Sin A = —7 — — — ) 5 n* -AX (x-X) f 71 whereA = — Verses 67-70 - Jya of bhuja and koti - Madhya graha substractecl from mandocca gives manda kendra and from sighrocca, it gives slghra kendra. From these quantities, bhuja and koti jya are calculated. In odd quadrant (visama pada), jya of passed arc is bhuja jya and remaining arc gives koti jya. Irt even (sama) quadrant, remaining arc gives bhuja jya, and passed arc gives koti jya. (Quotation of Surya siddhanta ends with verse 68). This means that if the kendra (manda or sighra) is less than 3 rasi (90°), then its jya is bhuja J 7 a * If li is between 3 to 6 rasi, then it is substraeted from 6 rasi. Jya of the balance arc is bhuja jya. 184 Siddhanta Darpana When kendra is between 6 to 9 rasi, we deduct 6 rasi from kendra. Jya of the balance arc is bhuja jya. If kendra is between 9 to 12 rasi, it is deducted from 12 rasi. Jya of balance arc is bhuja jya. If the quantity, from which deduction is to be done is smaller, then 1 rotation of 12 rasi (360°) is added to it. Notes (1) The rulers very simple and needs no explanation, if figure 12 is seen. ANBU is kranti vrtta in which U is ucca (manda or sighra). The planet moves in anti clock wise direction shown by arrow. Mi, M2, M 3 and M4 are positions of the planet in 1st, 2nd, 3rd and 4th quadrants from ucca position. Displacement of the planet along AB line is indicated by perpendiculars from M on AB i.e. parallel to NU line. Thus in quadrant 1, Mi,U is the passed arc of angle Mi,DU. Its sine is Mi, Yi, or OXi, called bhujajya. Similarly bhujajya in 2nd, 3rd, and 4th quadrants are OX 2 , OX 3/ OX4. . Kotijya is jya of complementary angle (90° - angle) or cosine in modern terms. It is indicated 8 Figure 12 - B4hu and Kotijya True Planets 185 by displacements along UN line. Kotijya in the quadrants are OYi, OY 2 , QY 3/ OY 4 . (2) Manda Kendra = Mandocca - Madhya graha. Sighra kendra = Sighrocca - Madhya graha Both are definitions and need no further comment. • Verses - 71-72 - Method of finding jya and utkrama jya of any angle - (Quotation from Surya siddhanta) From the chart we get sines and versines, only for angles which are multiples of 3° 45' (225'). Finding values for any intermediate angle is called interpolation.. Its method according to Surya siddhanta is — - — • . = Quotient (= completed parts of 225') + remainder (angle lapsed in next part) Jya of angle = Jya of previous part + remainder/225 x (Jya of current part - Jya of previous part.) Notes (1) This is simplest formula based on ratio and propotion i.e. rule of 3 (to find 4th unknown quantity). Difference in Jya of fractional part Angle of fractional part Difference in jya of completed part Angle of 1 part (225') * *£6 Siddhanta Darpana This assumes proportional variation of sine? difference. This is called linear variation or linear interpolation. (2) For small divisions of 3° 45' each; linear formula is sufficient to get accuracy upto a minute. If divisions are of 10° each, then sine difference doesn't increase proportionate to increase in angle. For such interpolation, Bhaskara II has given quadratic formula - y = yo +- (yi - yo) + ,' (72 - 2yi + yo ) where h = intervals (10') at which sines have been calculated; x is increase in angle; yo, yi, and V2 are difference of sine for successive parts. This formula was first stated by Brahmagupta. Vatesvara has given several forms of the formula. For R sines upto seconds, quadratic formula is needed. Brahmagupta expression for 225' intervals is - R sin (225't+0') - - - (where 0' < 225', t is an integer) - sum of t R sine - differences ej_ r t th R sin diff + (t + l)th R sin diff 225 L .2 6 ' t th R sin diff + (t + l)th R sin diff -^5~ ^2~ ~ 1(1) = sum of R sine differences + ^ (t + 1)* R sin difference + -r 1 0' {&' 2 225 diff.] 225 — 1 [ ( t+1) th R sin diff. - t th R sin (2) (2) is equivalent to quadratic formula. 187 True Planets (3) Madhava's formula - If t is a positive integer and < 225', then R sin (225' t + 0) = sum of t R sin - diff 6 x [R cos 225 (t + 1) + R cos (225 t)] + 2R This has been quoted by NHakantha in his commentary on Aryabhatiya. Verses 73-74 : Finding arc from any given jya - We substract the greatest jya pinda/ smaUer than given jya, from the given jya. The difference of jya is divided by difference of the jya pinda substracted and next bigger jya pinda. It is multiplied, by 225' and result is added to the completed arc. Notes - (1) This is again linear interpolation whose proof is similar to reverse process of finding sine of a required arc. (2) There is a similar quadiratic formula for this process also. (3) Geometrical proof of the formula (linear and quadratic) — Figure 13 shows the graha of R sine with its arc (or angle) On OX axis xo, xi x 2 are points indicating angles or arcs where value of sine is .XX, x, Figure 13 - Finding sine or arc for intermediate values 188 Siddhanta Darpana known. R sine at these points is no, n t , n 2 . We have to find value of R sine x = D on the graph when Ox is known in interval (1,2) Alternatively, if D is known, we have to find x point. In linear interpolation, we assume that value of R sine changes along the straight line AC from no to ni. For small intervals it is almost same as curve AC; Dx is perpendicular on X axis cutting AC at D' and AB at E. Thus D' is a good approximation (linear) for D on the curve. Now D'E = D'x - Ax = increase in value of R sine for desired angle. C B = increase in Rsine over interval (0, 1) " D'E - AE HenCe lc-= AH"'.., When x is known, we know AE, AB = 225' and BC is already known (value of R sines). Then fourth quantity D'E can be known. Required sine is D'x = D'E + A x which gives the formula. D ' E x AB Similarly Xqx = AE = — - can be known and O x = Ox + xxo * . Quadratic interpolation - ni-rto is difference between the interval. (n 2 -ni) is difference in next interval. Average rate of change is (ni - n 0) + (n 2 - ni) 2 Increase in rate of change within the interval AE (m - no) - (w2 - «i) " AB 2 True Planets 189 Second term is the extra term for quadratic form of Brahmagupta. Verse 75 : Relation between sine and cosine , From square of radius, deduct the square of jya, take square root of the quantity* Result will be jya of koti of the angle (3 rasi - angle) or kotijya. Similarly, jya can be calculated from kotijya. Note - Cos (9O°-0) = Sin 6 or sin (9O°-0) = CosO Sin 2 6 >* Cos z fl = 1 Hence V r2 _ R 2 sin 2 e s R Cos 6 Verses 76 to 88 : True motion of mandocca by difference from parocca - Mandocca of mahgala, budha and sani and sighrocca of budha move both forward and backwards as observed by me (author). To find their true motion another entity (devata) called 'Parocca' has been assumed which affects these points (verse 76). Parocca of mahgala (its mandocca) is at 6 rasi (180°) difference from madhyama surya. Parocca of budha mandocca is its Sighrocca. Likewise, man- docca of budha is parocca of budha's sighrocca. Parocca of sard's mandocca lies at 18° less from madhyama sani. Parocca also is an invisible form of Kala like mandocca. Similarly there are many small planets in the sky. Around planets, satelites move in circular orbits. Motion of these small planets and satellites can be seen only with instruments (telescope). They also move from west to east. They have not been described in Brahma and surya siddhanta, hence they are not been explained here. * igO Siddhanta Darpana Substract parocca from iighrocca and mandoc- ca of budha and from mandocca of sani. R sine of the resulting angle is multiplied by 680 for budha and by 300 for sani. Product is divided by radius (3438). Result will be in lipta. If Para kendra (distance of manda or sighra from Parocca) is between 0° to 180° /it is added in mandocca and for 180° to 360% it is substracted. For slghrocca of budha, its opposite procedure is followed (substrac- tion for parakendra in 0° to 180° and addition other wise. If sighra kendra of mangala is in sue rasi's starting from makara (i.e. from 270° to 90 V then sphuta sanskara is not needed for its mandocca. If sighra kendra is in six rasis beginning with karka (i.e. 90° to 270°) then mean mandocca will be corrected to find the true value. Mandocca of mangala for previous year (almost same for current year) is substracted from its parocca (mean smv + 180°). Result is its para-kendra. R sine of para kendra is multiplied by 450 and divided by 3438 = radius. Result is multiplied by kotijya of first sighra kendra in kala and then divided by kala of 3 rasi (5400). As before, result is added in 1st six rasis of para kendra and substracted in other rasis from mandocca of mangala. For daily motion of mandocca - Substract mandocca from parocca of mangala (i.e. 180° + mean sun). Its kotijya is multiplied by 450 and divided by radius (3438). This will be kotiphala for correction of mandocca speed. If parakendra is in 0" to 180°, kotiphala is added to sighra kotijya and substracted for parakendra between 180° to 360°. Result is multiplied by mean daily motion of sun True Planets 191 # ■ and divided by radius (3438). Result will be daily motion of mangala mandocca due to its parocca. Similarly for sani, find kotijya of its difference f parocca and mandocca. Multiply it by 300 and divide by radius 3438. Result is kotiphala for correction. Kotiphala is multiplied by mean daily motion of sani and divided by radius 3438. This will be daily motion of sani mandocca as corrected for its parocca effect. For budha, Kotijya of its para kendra is multiplied by 680 and divided by radius 3438. Result will be Kotiphala for correction. Multiply this Kotiphala by daily mean motion of budha sighrocca and divide by radius 3438. This will be daily motion of true budha sighrocca. Correction in mandocca speed is addition for parocca kendra in 1st and 4th quodrant (270° -90°) and otherwise deducted. Correction in sighrocca is in opposite manner. Notes - (1) Author has not given reasons for such correction. His mention of observation of small planet by telescope indicates that these correction are based on some modern charts like Le - verrier's chart of 1850 or some nautical almanc available in his time. He has clearly mentioned that these have not been discussed in other siddhantas which indicates his corrections are adopted from some almanc or results of telescopic observation. (2) It is difficult to guess as to what correction was sought to be achieved by these methods. However, mathematical form of these formula will indicate the reasons of these corrections. *92 Siddhdnta Darpana (a) Definition - Parocca' is a mathematical | point in ecliptic from which deviation in mandocca I of mangala, budha and sani and sighrocca of budha \ can be calculated. (b) Parakendra (P) = Sighrocca or mandocca j - parocca. Parocca of Budha sighra = mandoca of budha Parocca of budha mandocca = sighrocca of budha Parocca of mangala mandocca = madhyama ravi + 180° Parocca of sani mandocca = madhyama sani - 18° (c) Madhyama mandocca of all the planets is given in madhyamadhikara. Sighra of inner planets budha and sukra are the planets themselves. Sighra j of outer planets mangala, guru, sani is mean sun. 4 Thus the tara graha, affected by own orbit as well as earth orbit (or relative motion of sun) have 1 sighrocca as the planet of smaller orbit (and hence of faster rotation). (d) Para kendra of Budha sighrocca (Bi) i.e. PBi, = Bi - B 2 where B 2 is mandocca of Budha. Para Kendra of B 2 (mandocca of Budha) PB 2 = B 2 - Bi Correction in mean Bi = 680' sin (Bi-B 2 ) Correction in mean B 2 = 680' sin (B 2 -Bi) Sin (Bi-B 2 ) is positive when PBi = B r B 2 is j between 0* and 180° then it is negative correction J For mandocca it is opposite. Hence for both budha | mandocca and sighrocca, correction is ■i True Pla^ s 193 680' Sin (B 2 -Ba) - - - (1) Parakendra of sani mandocca, Ps is S 2 - (madhyam Sani - 18*), S2 = mandocca of Sani or Ps = S2 + 18* - madhya sani = S 2 + 18 - S say - - - Correction to mandocca is 300' Sin Ps - - (2) Since it is to be added when Ps is between 0° to 180° i.e. Sin Ps is + ve, the formula indicates correct sign. Sighra kendra of mangala = mean sun - mean mangala. Skm = S m - M When S Km is between 270° to 90% no correction is required. Correction to mandocca is done only when S Km is between 90° to 270° i.e. earth is on same side of sun as mangala. Amount of correction - Parakendra of mangala Pm = (Sm + 180").- M2 where M2 is mandocca of mangala (This is opposite substraction of the earlier process) Correction in mangala mandocca 450 sin Pm x cos S Km 5400 Sin Pm is + ve for Pm between 0° to 180° a nd it is added to mean mandocca. Speed of mandocca is obtained by obtaining the differential coefficient of the corrections. Position of true mandocca of mangala # 194 Siddhanta Darpan 450 sin Pm x cos S Km 5400 450 cos Pm x cos S Km d __ * -r~ rm 5400 dt = speed of mean sun. Speed s dPm _ dSm IT ~ dt Thus speed of mangala mandocca >peed 450 cos Pm . cos S Km 5400 -(fl Speed of Budha Slghra or Mandocca = 680' Cos (B 2 -Bi) - - - (5) Speed of Sard mandocca = 300' cos Ps (6) (3) Reason and assumptions of these corr tions - (a) Sard mandocca motion - Motion of mandocca cannot be observed in a life time or ev in a thousand years because it rotates only 39 tir in a kalpa. Hence it is not oscillatory motion mandocca which could be observed by the autho* This appears to be correction due to effect of guru' attraction on Sani motion. At the time epoch his observation after 1869 AD, guru was behin Stni for 5-6 years. Hence parocca of sani has bee assumed to be slightly less (18°) than mean sani (b) Mangala is corrected, only when it influenced by earth when both are on same sid of sun. Hence this correction in mandocca is t account for influence of earth. (c) Correction in sighrocca and mandocca Budha is to make correction of elliptic orbit Budha slghra (i.e. Budha itself) and its hi inclination with sun's ecliptic; 7° as seen from s True Planets 135 These are reasonable assumptions of the origin. It needs further research and verification, gut obviously, these corrections tallied with obser- vations in author's time. Verses 90-103 - Manda and sighra Paridhi - For any planet, attraction by its mandocca is multiplied by kala of a circle (21,600) and divided > by radius (3438). Result is manda paridhi. Maxi- mum mandaphala varies with manda paridhi. (90). Sighra paridhi of mangala, guru and sani is more in even quadrants compared to odd quadrants end. For budha and sukra it is opposite. (91) According to Surya siddhanta - Manda paridhi of ravi is 14° in even quadrant and 13'4Q' in odd end. Similarly, mandapraidhi of candra is 32* in even quadrant and 20' less in odd quadrant i.e. 3r40\ (92) According to Siddhanta Siromani of Bhaskaracarya, mandaparidhis of ravi and candra are constant and are 13 e 40 / and 31° 16' respectively. (93) I (author) have calculated the values of manda paridhis of ravi and candra by observing conjunc- tion of moon with stars and difference in rasi of moon and sun (phases of moon) (94) In odd quadrants, mandaparidhi of sun is 12° 6' and of candra is 31° 30' (95) If manda kendra is at end of 4th quadrant, mandaparidhi of ravi is 11*30'. Now, method for finding manda paridhi at other places is being told (96) 196 SiddMnta Darpana -- - . * Multiply kotijya of ravi manda kendra by 6 and dKde by radius (3438). Result wffl be in kala ete If manda kendra is in 1st or 4th quadrant, tSJLt this result from 18' ^ At other positions this will be added to 18 kala. (97) Either of the results is multiplied I by R cos of ^Snd^Idhi afendtf od^adrant (IT*. wSn mandakendra is in 2nd or 4th quadrant. In other quadrants it is added (98) This method is adopted for accurate calcuk- tion For rough work, l/9th part of previous result Padded. This will give accurate results only on parva sandhi (Purnima or amavasya). Kotijya of candra manda kendra (R cosine) » multiplied by 30 and divided by radius (3438). SSttaK* etc is added to the manda paridh afend of odd quadrants &'*>>.***,££ kendra is in six rails starting from karka. In other PosSns, it is subtracted to find sphuta manda P Manda paridMs of planets are -mangala 69- budha 27% guru 34*30' sukra 12 and sam 39 Sighra paridhis at end of even and odd quadrant ** tant 237* Mangala 238' Budha 139° Guru 70 ^ Sukra 26 1 * Sani 39- * True Planets 197 Notes (1) Mandaparidhi is correction method for elliptical orbit which have been assumed circular for first approximation. (2) Sun and moon are directly in an ellipse around earth. But other planets take their position aS a result of two orbits ~ orbit of sun around earth (apparent) and orbit of planet round sun. Correction from mean position due to smaller of these orbits is done through sighra paridhi. (3) The method of correction by manda and sighra paridhis will be explained after calculation of these for tara grahas. Verses 104 to 112 - Sphuta manda and sighra paridhis for tara grahas - Difference (of 1") between sighra paridhis at the end of odd and even quadrants is multiplied by bhuja jya of sighra kendra and divided by radius (3438) i.e. multiplied by sine of sighra kendra. This result is added to the sighra paridhi at end of previous quadrant, if it is rising in current quadrant. Otherwise/ it is substracted. For more accurate value of sphuta sighra paridhi of mangala, we add 1/30 part of bhuja kala of sighra kendra (i.e. R sine of sighra kendra is minutes). Manda paridhi of mangala is 69° only at the end of quadrant. To find the intermediate values, we select the lesser part of manda kendra - among lapsed part and remaining part in the quadrant. R sine of that angle is multiplied by 8" when manda kendra is in six rasis starting from karka, or by 4° w hen manda kendra is in 6 rasis starting from ^akara. Result is divided by R sine of 1-1/2 *rasi 19S Darpana (45°) = 2431. .Result is converted to degrees etc and added to 69° which gives sphuta manda paridhi of mangala (at any place). When manda kendra of mangala is between 4868' and (4868' + 1590') or between (15,142') and (15,142'+1590'), its mandaparidhi is taken as equal to its mandaphala of 3 rasi i.e. 11 2'47". We find the lesser of lapsed and remaining^ parts in quadrant of Budha manda kendra. Its R sine (jya) is divided by 9 and result is substracted from manda paridhi (27°). We get sphuta manda paridhi. R sine of sukra manda kendra is multiplied by 2 and divided by radius R = 3438. Result in- degrees is substracted from manda paridhi (12°) to find its sphuta value. Notes - (1) As first approximation, planetary orbit is considered circle with earth at centre (for moon and sun). M :i Figure 14 - Epicycle (Niocca vrtta) *\ '■ 'A -A True Planets 199 Figure 15 • Eccentric Circle But orbit is elliptical with earth at focus which is away from centre at a distance of ae towards apogee (mandocca). Here a = semi major axis and e is eccentricity of ellipse. At first step of approximation, we shift the circular orbit by distance ae in direction of major axis (Fig. 15). YU X N is orbit of mean planet whose apogee is U and V is the point of mesa 0°, vernal equinox, from where longitude is measured. Another circle with centre at C in direction EU is drawn with same radius. In both the circles the planet rotates with same speed. Thus every point on the eccentric orbit (Prati vrtta) PUi L is at a distance CE in direction of U. At apogee, mean planet is at U and true planet at Ui. When mean planet is at M, corresponding planet on prativrtta is at Ti where MTi = EC and both are parallel. Same displacement can be done by assuming movement of spasta graha (true planet) on another 200 Siddhanta Darpana small circle whose centre is on madhya graha. The circle is called manda paridhi (epiciycle) which rotates fixed with radius vector of mean planet. However, movement of spasta graha on manda paridhi is in opposite direction to the motion of madhya graha but with equal angular speed. At apogee position in Fig 14, mean graha is at U and true graha is at Ui in same direction, When mean planet moves to M in anti clockwise direction by angle 6 = Z.UEM (manda kendra), the true planet moves by same angle = TjBBi in opposite direction. Thus Ti is always in direction of mandocca i.e. MTi is parallel and equal to CE. Thus by construction of manda-paridhi also, all points of madhya graha orbit are shifted by distance EC in direction of EU towards ucca. Thus both the constructions are equivalent. (2) Ellipse is symmetrical with respect to centre but not from focus which is centre of true orbit. Next step of approximation to make it toally equivalent to elliptical orbit is by changing the radius (or equivalently circumference) of manda paridhi at different places. Let E be origin and EU direction of X axis, EX being direction of y axis. Radius of mean orbit (deferent) EM = R Radius of manda paridhi for manda kendra 9 = m + n cos 6 n has lowest value at 90° or 270" when cos 6 = m+n has highest value at apogee (0=0 e ) or at 6 = 180° (-ve) Coordinates of point Ti are - x = EM, cos $ + MTi - - in direction of EU -:« True Planets 201 = R cos + (m+n cos 6) or x - m = (R+n) cos 6 - a cos y = Rsin0 = bsin0 where a = R+n, b = R are the semi major and semi minor axis. This is parametric equation of ellipse with centre at (m,o) i.e. at distance EC=m from centre of kaksa vrtta towards mandocca. From this, e - V^P = V(R+n) z -R* = V 2T1 R + r, 2 e - va — _d_ R + n R + n a 360* 20* 1° , . ..-.■ , R = ^— , n = — = (surya siddhanta) 2 Jt 2 jt 3x2^ Hence V -.360 + ^ ^ ?_ = V & x 360 + 1 e 360 + V^ 1081 = 0.043 (real value for sun is 0.0167) Geometric equivalent of the correction - Without continuous varying the radius of mandaparidhi also, we can obtain the true position of planet. In figure 14, join CT, which cuts deferent (orbit of mean planet) at S. Produce ES and MTi to meet at T. Then MT is the radius of true epicycle at M and T is true position of sun. Similar construction can be made for prati- vrtta (eccentric circle) also in figure 15. CTi cuts deferent at S and ES cuts MTi produced at T which is true position of planet. (3) Bhujaphala is equal to equation of centre- In figure 14, Sun's mean longitude = arc VUM 202 Siddhdnta Darparut True longitude = arc VUS Difference between the two, i.e. arc SM, is the sun's equation of centre MA is perp. to EU and T1B1 and SB be perpendiculars to EM or EM produced. T1B1 is called bhuja phala or bahuphala and BiM is called kotiphala. A 5 BiMTi and MAE are similar. Then Ti Bi .MA Ti M ■ ~ EM or TiBi, i.e. sun's bhujaphala TiMxMA EM 14° TiM = radius of epicycle (mean) = - — MA = R Sin 0, EM=R Hence, Bahuphala 14° Sin 6 2 Jt 14 X 6C Sin© In = 133/7 Sin = 0.388 sin 6 radians. With mean value of manda paridhi 11 "48', it is 0.327 sin 6 which compares well with the modern value of 0.334 SinO. Plotemy had given 0.416 sin 6 radians. (4) In eccentric circle, geometric construction gives the method of successive approximation described later on while dealing with true speed. (5) Heliocentric amomaly through Sighra kendra. —Position of tara grhaas depends on two orbits - apparent orbit of sun round earth and orbit 203 True Planets f planet around sun. Smaller of the orbits is called lighra paridhi and madhya graha corresponding to average motion is bigger orbit. Figure 16 -A' Figure 17 Figure 18 Like correction of elliptic orbit through manda t paridhi, correction from heliocentric to geocentric position is done through sighra paridhi. It can be an epicycle, or eccentric circle, as shown in figure 18. 204 Siddhanta Darpana It is to be proved that sighra kendra is same as heliocentric anomady and sighra phala is conversion from heliocentric to geocentric position. This will show correctness of sighraparidhi method. Figures 16 and 17 show the anomalies with sun as centre. First figure is for inner planets venus and mercury, marked as V. Second figure is for superior planets mars, jupiter and saturn, marked as J. Figure 18 is for sighra paridhi - both for inferior and superior planets. Figure 16 and 17 - S = Sun, E = Earth, SA = direction to mesa 0°, EA' = direction to 0°. SV, EV are heliocentric and geocentric directions of sighrocca. V is actual planet and V is imaginary point (sighrocca of inferior planets) Draw EJ' I I SJ. Radius of inner and outer circles are r and R. K is radius vector to the planet (from earth). Figure 18 - Ei = Earth's centre, E 2 - Centre of eccentric circle (It will be proved as centre of sun) Mi, M2 are mean planets in deferent (kaksa vrtta) and eccentric (Prati Vrtta). E a E 2 * r = antya phala jya. R is radius of both circles. K is radius vector to planet known as sighra karna. Sighra kendra (anomaly) = Longiude of sighrocca - Longitude of planet (fig. 18) = L aEi Ai - Za E1M1 = Lb! E 2 Ai - Za' E 2 M 2 = m In Fig 16, ZA' EVi is longitude of sighrocca, ZA'ES is longitude of sun treated as madhya graha of inferior planet. sighra anomaly, m = ZVSS' = ZV'ES True Planets 205 = ZA'EV - ZA'ES In fig. 17, m = L S' SJ=/L SEJ = L A'ES-iiA'EJ' = L A'ES-ii ASJ = Longitude of Sun (treated as sighrocca of superior planet) - longitude of planet from Sun known as mandasphuta graha. = Sighra kendra Consider A s ESV, JSE and E1M1M2 of the three figures. L ESV = 2LJSE = L Ei M1M2 = 180* - m If value of sighra paridhi is taken such that SV SE M1M2 — = — = (1) SE SJ Ei Mi K) all the three triangles will be similar. Thus sighra kendra is same as heliocentric anomaly and sighraphala L MiE M2 = L SEV = L SJE. K 2 = R 2 +r* + 2 Rr cos m (2) Comparison of values of orbit known in modern astronomy , shows that value of sighra paridhis have been chosen correctly, so that equation (1) holds - . Sighra paridhi small orbit (radius or circum) — ' 360° "~ Larger orbit ' Hanet Sighra Deferent Ratio Value in*modern paridhi astronomy (average) earth = 1 Mercury 139.5 360 .3875 0.387 Venus 261.5 360 .726 0.723 Mars 237.5 360 1.519 1.52 206 Siddhanta Darpam Jupiter 69.5 360 5.18 5.20 Saturn 38.5 360 9.35 9.5 (6) Formula for sphuta paridhi - Difference between values at end of odd and even quadrants is 1 . Hence if is angular difference from lowest position, addition will be Sin 6 in degrees. For further correction in mangala 1/30 part is . 31 sin 6 --rt-oo o- a added i.e. correction is = 1.033 bin v Mangala manda paridhi is minimum in ends of quadrants (90° interval) and it is maximum in between (45° from ends). Difference from minimum 69° is 8° in 90° to 270" and 4° in other half. Hence correction in 90° to 270" fe- 8° x sin 6 4° sin 6 + in other half it is r— ; — — R sin 45° R sin 45° It is constant in two intervals 4868' and 4868' + 1590' and (15,142 to 15,142' + 1590'). Then mandaparidhi is 11°2'47'. Budha manda kendra = lesser interval from ends of quadrant. R' Sphuta paridhi = 27° - — sin $ Sukra manda kendra = Sphuta manda paridhi = 12° - 2° sin 6 (7) For outer planets, earth is on same side of sun and closer to # planet for sighra kendra ift even quadrants (closest at end). So its sighra paridhi is more. For inner planets, it is opposite. ,3 .1 True Planets 207 Minute chanes in slghra paridhi are due to eccentricity of slghra orbit also. (8) Bhaskara II, has measured difference of mandocca - madhya graha in anti clockwise (position direction) and madhya - sighrocca in opposite direction, Madhya is faster than mandocca but slower than sighrocca. However, both measured same way make no difference. Verses 113 to 120 - Bhuja and koti phala and karna According to surya siddhanta, sphuta manda paridhi multiplied by R sin of bhuja of manda kendra and divided by 360* gives bhuja phala. When this paridhi is multiplied by kotijya (R cosine) of bhuja and divided by 360° it gives kotiphala. (113) When arc is smaller than 225', it is same as its R sine (jya). Then arc or R sine need not be converted to each other. (They are taken equal). Only when arc is more than 225', its sine is to be calculated. According to surya siddhanta, slghra kotiphala is added to trijya (3438) when slghra kendra is in six rasis beginning with makara (i.e. 270° to 90°). For other sighra kendras (i.e. 90° to 270°) it is substracted from trijya. This is kotija bhujaphala, used for correction °f radius and should not be considered an arc. In surya siddhanta - Squares of bhuja and koti phala are added and square root of sum is taken. Then we get slghra karna. Bhuja phala Multiplied by trijya (3438) and divided by slghra karna gives sighraphala in minutes of arc. Slghra 208 Siddhattta Darpatui phala is used for first and fourth corrections o| five star planets starting from manga la. Sun and moon become spasta with only one correction with mandaphala. But in five tara grahas, sighra phala correction is done at first, then mandaphali correction is done twice. At fourth step, sighra phala correction is done again. When sighra kendra or manda kendra is less than 6 rasi, sighra of manda phala is positive, hence always added for correction. When kendra is more than 6 rasi, phala is substracted. Notes : (1) List of given formulas Manda Paridhi __ Mandatrijya 360° " Trijya(3438) Hence manda trijya r, bhuja of manda kendra give m, • u i r • R sin ° • a Bhujapnala = — = r sin 6 In sighra paridhi, R + r cos $ is calculated for known distance of true planet. Cos 6 is positive from 270° to 90° hence it is added, otherwise substracted. Stghra Kama = Vfihuja Phala 2 + Kotiphala 2 (both o f sighra paridhi) i.e. r * Vj* s j n 2 q + j2 cos 2 £ = radius of sighra paridhi or sighra karna, 6 = bhuja of sighra kendra. c- u u i • a r • R sin Sighraphala = r sin 6 - Proofs are obvious when diagram of sighra or mandaparidhi is seen. Sin is positive when B is between 0° to i80° hence manda or sighra phal* is positive and is added. True Planets 209 Verses 121 - 123 - Correction in madhya tara graha — Madhya graha corrected by half of sighra phala (addition or substraction) gives first graha (corrected). Manda kendra is calculated for first eraha and half of its mandaphala correction gives second graha. Manda kendra is again calculated for second graha. Its correction by mandaphala (full) gives third graha. For third graha, sighra kendra and sighra phala is calculated. On correction of third by this sighraphala, we get fourth graha which is the true position of planet (spasta graha). Notes (1) - If madhya graha is Po^ 1st and 4th sighraphala are Si, and S4, 2nd and 3rd mandaphala are M2 and M3, graha after 1st, 2nd, 3rd, 4th correction are Pi, P 2 , P3, P4 then Pi = Po ± J M 2 P 2 = Pi.±Y i P 3 = P 2 ± M3 P4 = P 3 ± S4 Here Si and S4 are calculated for Po and P3, and M 2 and M3 are calculated for Pi and P 2 . Correction order can be indicated by S M w ■ - + y + M + s (2) Aryabhata method For superior planets 2 + y + M + s For inferior planets (vernus and mercury) 220 Siddhanta Darpana S - + M + S (only 3 steps) He has calculated slghra kendra in opposite direction (Sighrocca - planet), hence it is substracted for 0° to 180°. M S Bhaskara I method — + — + M + S 2 2 S For inferior planets — + M + S (only 3 steps) t S/2 for inferior planets is corrected in reverse way and sfghra kendra is calculated from its mandocca. Surya siddhanta method is the traditional and most popular method in country. It has been! followed in siddhanta darpana also. 1 (3) Further explaination of variations in manda ^ and slghra paridhi. (In continuation of note 2 after j verse 112) I 772 - Eccentricity e of orbit = — I a i where m smallest value of manda paridhi 1 a = semi major axis = R+n I I n = difference between maximum and mini^ mum values of manda paridhi. Thus e = :r = — approx, n very small. R + n R rr J It is also given by e = V2r ^ R + n2 R + n * V r 2ft approx as n is very small. R True Planets 211 Thus approximately, e = — = V^n R R or m = V2«R m or n = — , m = eR This gives method of calculating maximum manda pardhi and its correction term For sun, max. paridhi is 14° (= 2 Jim) and max correction is 20' (= 2#n) e = v^ = 0.043 R m e = 039 K This is similar by both method. Thus correc- tion depends on value of max manda paridhi. (4) Reasons for starting correction with sighra phala - Mandaparidhi is measure of eccentricity of orbit (e=m/R) which is very small and less than 1/50. Shighra paridhi is ratio of smaller orbit to bigger orbit among the orbits of earth and planet round the sun. This varies from 1/9 to 3/4 approximately. Hence at first step manda correction can be neglected and only sighra correction in done. For inferior planets manda correction also is done in sun's orbit, not in the orbit of planets. Hence alternatively, manda correction can be done before sighra as stated by Aryabhata and Bhaskaral. We do not calculate manda or sighra kendra from true planet, but from mean planet which is an approximation. Hence only half corrections are done for sighra and manda in beginning. Prob- 212 Siddhdnta Darpatut ability of negative or positive error will be both equal in half corrections and are likely to cancel each other. Then manda correction in full gives heliocentric anomaly of the planet - called manda spasta graha. Its last correction by sighra phala has been explained in note (3) after verse 112. Since sighra and manda corrections are com- parable, their half correction only is taken at a time. After 2nd correction error is reduced and after full manda correction, exact sighra phala can be determined. (5) This is a type of calculation based on probabilistic value of errors which is called 'Monte- Carlo method' in modern numerical analysis. Reduction of error at each step is similar to 'iteration method' for system of non-linear equa- tions. "o Xt Xg X Figure 19 - O<0*<1 *» X 2 X x t Figure 20 for - 1 <0'<O Method of iteration for numerical solution. Solution for y = <D(x) is its point of intersection 711 True Planets with line y = x whose slope with x axis is 1. Figure 19 explains the approximations when slope is positive and figure 20 indicates negative slope of y = 4>(x) Slope is <*>' (x) or — , it is positive when function in increasing, negative when it is decreas- ing. In both cases its numerical value is less than 1 i.e. slope of y= x. Only in such case successive approximations will reduce the errors at each step. For slghra and manda corrections also, the correc- tions are much smaller than 1 as explained in previous para. Xo is the first approximation (like madhya graha). x a , x 2 , x 3 are next approximations. When function (slghra phala or manda phala) is increasing, i.e. correction is additive, all the approximations are on left side of, or less than true value x. When correction is negative, i.e. function is decreasing (Fig. 10) xi, x 2 , x 3 - - - alternate on either side of the true value. In both cases, diagram shows that errors decrease at each step, which was purpose of our corrections. (6) Reasons of half corrections in first two steps - By full correction we may over correct and may not decrease the error which is required for iteration. Half correction will always reduce the error. Full investigation can be done only # with Lyapunov's conditions of stability. However taking half of the approx value of correction, probability of positive and negative error both are same and it wil be approaching zero in end. 214 Siddhanta Darpana It is. similar to methods used by computer which divide the line segment into two parts for Y X. b s b-bo Figure 21 numerical approximation. In figure 21, solution of f (x) = is its intersection with x axis. On one side of true x*, there is a point a Q for which f(x) is - ve and on other side f(b c ) is + ve. We take midpoint c of interval (a G , b ). If f (c) is negative, we make it the new point in place of a, where it is negative. Thus we go on dividing the interval for better approximations. Verses 124-131 - Special correction for mangala and budha - For mangala, 3rd and 4th phalas in kala are muthplied and divided by 10. Result is substracted from last karna. Then we get result in lipta etc. This result is added to 4th (sphuta) graha when manda kendra is 0" to 180 °, otherwise substracted. If 3rd kendra (manda) of mangala is from 90° to 270°, 4th phala is substracted from 55, result is multiplied by manda koti phala of 3rd operation. This result is substracted from 4th karna. This is substracted from 5th graha, then we get 6th sphuta graha. If manda kendra of 3rd planet is 270° 'to 90° then this correction is unnecessary, (fifth graha will be true). Madhyama budha is substracted from budha sfghrocca, already corrected for parocca* This slghra kendra is used to find half of first slghra phala, True Planets 215 which is kept in 1st place. In 2nd and 3rd places, we keep half mandaphala obtained from madhyama budha after 2nd operation (correction with half manda phala). At 2nd place, this mandaphala is multiplied by half sighra phala at 1st place and divided by half of the 4th sighra phala. 1/3 of the result is substracted from mandaphala at 3rd place. The new manda phala is used to make 3rd correction of madhyama budha. From that 4th sighra phala is obtained and kept in 2 places. At 2nd place it is multiplied by 3rd kotiphala divided by radius (3438)). Result is added or substracted at 1st place (addition is done when manda kendra is 90° to 270°). This sighraphala is used for 4th correction. Then we get more correct result com- pared to Surya siddhanta. Notes : The rules are lengthy and confusing when stated in words. (1) Rules for mangala - Po, Pi, P2, P3, P4 are the mean planet and the planets after 1st, 2nd, 3rd and 4th correction. Si, S 4 are sighra phala for 1st and 4th corrections, when sighra kendra is calculated for Po and P3. M 2 , M 3 are mandaphala for 2nd and 3rd correction where mandaphala is calculated from manda kendra of Pi and P 2 . # Thus Pi = Po + Si/2 (S & M may be + ve or - ve) M2 P2 = Pi + — , P3 « P2 + M3, P4 = P3 + S 4 ^ Si Mz „ „ . Thus P4 = Po + -z- + -r- + M3 + S4 = True graha r4 __ — — J n liptag 216 Siddhanta Darpana r\, r 4 are sighra radius for Si, S4 and ri, r 3 manda radius for M2, M3. If is manda or Sighra kendra (bhuja), M or S = r Sin 6 For mangala we obtain P 5 and Pe and further corrections of true planet. f M3XS4 x Fifth correction X5 = P5 = P4 + X5 when manda kendra of P4 is between 0° to 180° or = P 4 - x 5 when it is between 180° to 360°. When manda kendra of P3 is between 270° to 90° this P5 is the last correction needed. If manda kendra of P3 is between 90° to 270 - , sixth correction xe = r 4 - (55 - S 4 ) r 3 cos 3 P 6 = Ps - xe (2) Correction for Budha - S' is sighra phala Si of budha corrected for parocca. From — we calculate M2'. For third correction we do not calculate Nfe' from 2nd planet. / c ' \ 1- S1 3rd correction = M2 = X 3 3S 4 S 4 is calculated by general method. The new sighra phala after P 2 + x 3 = F3, i&| called S 4 '. Fourth correction x± = S 4 ' (1 ± r 3 Cos ^3) Addition is done when 3 is 90° to 270° P 4 ' = P 3 ' ± xa (addition for 0° to 180°) True Planets 21? P 3 ' and P4' are planets obtained by revised method. Verses 132-138 - True speeds of sun and moon. Now true speed of graha is considered. The speed changes every moment, but sphuta gati of a day is the difference between sphuta graha on two successive days. Strictly this will be average daily speed for that day. Dainika gati of mandocca, substracted from dainika gati of mean graha, gives danika gati of manda kendra. Dainika gati of manda kendra multiplied by *manda kotiphala and divided by radius gives manda gatiphala for one day. This is added for manda kendra in 6 rasis from karka ; in madhya gati of graha. Other wise it is substracted from madhya gati. This result will be manda sphuta gati for one day i.e. from sunrise to the next sunrise. At sunrise, difference of true moon and true sun gives the balance part of current tithi. This (added to sunrise time) gives ending time of tithi. At that time true moon is again calculated and further correction of tithi end time is done. This accuracy in knowing beginning and end of tithi is needed only for ascertaining time of eclipse or oi sraddha (last rites). For normal works, the # true position of sun and moon and their speeds al sunrise will be assumed constant for the day. Notes : (1) List of all terms as revision and summary - Mandocca-Madhya graha = manda kendra M. 218 Siddhanta Darpana * Sighrocca - madhya graha = Sighra kendra S Manda kendra or Sighra Kendra = 6 Figure 24 True position of graha is P' while mean graha is at P. Radius r of manda or sighra paridhi is PP. EP' is karna = K (Sighra or manda) EP = R, radius taken a§ 3438'. 2 n r is expressed in degrees of manda or sighra paridhi. PN' is perpendicular on Karna EP', N' is true position on Kranti Vrtta. Thus PN' is the correction in mean motion called sighra or manda phala. Mandaphala = PN' = almost P'N. It is slightly less than P'N, perpendicular from P' to EP extended. P'N = Doh phala or Doh jya = r Sin w j . i t>v„ r sin x R Mandaphala PN =— r R + r cos 6 Kotiphala PN is addition to the mean trijya in that direction PN == r cos 6 Sighra or manda karna K = EP' ■1 True Planets 219 K 2 = (R + r cos 6) 1 + (r sin Of = R 2 + r 2 + 2 r R cos 6 (2) Now the speed can be calculated with help of differential calculus. These results cannot come by any other method and are according to surya siddhanta. In figure 24 in above para, <S> is angle of ucca point U with mesa 0° at A. Then madhyagraha at P = <£ + 0, mandocca = dP dO d Thus d7 = ^r + IT (t 1S tune) or dainika gati of madhya graha = gati of mandocca + gati of manda kendra. True graha is at N' = P + r sin (negative correction) dN' dP a d0 or = — + r cos — as r is constant. dt dt at Thus additional gati i.e. correction == r cos .e, d6 dt = kotiphala x gati of manda kendra. Verses 139-142 - Sighragati of tara graha. Sighrocca gati —madhya graha gati = Sighra kendra gati. Sighra phala is substracted from 90 °, it is multiplied by daily motion of sighra kendra and divided by sighra karna. Result substracted from sighrocca gati is sighra sphuta gati. If it is nu>re than sighrocca gati, reverse substraction gives retrograde motion. In this way .5 tara graha have two types of gati — manda sphuta and sighra 220 Siddhanta Darpana sphuta. Ravi and candra have only manda sphuta gati. Notes : (1) Like above, madhya graha (manda sphuta for sighra gati) is at P, and P' is sphuta graha. Sphuta kendra O-P is given by O where O is longitude of sighrocca. dO - dP dO "77 = j. + "TT/ t is time measured in days i.e. dainika gati of sighra kendra = gati of sighrocca - gati of madhya graha P is sphuta, its component perp to radius is 4 P + r sin $ dP' dP n d0 Hence -3- = — + r cos — at at at dO d$ „ d$ = —rr - "TT + r cos0 . t~ dt dt dt dO d0 = dT"& (1 " rc09fl > dP' d O d ( R- Rr Cos \ R dt A d * Thus negative sighra gati phala is R - r . R cos R sin 90° - Kotiphala of sighra sighra Karna v PP / (2) Exact derivation assuming variation erf karna also - True Planets 221 Figure 25 Figure 26 (Inferior planet like venus, mercury) superior planet E = Earth, S = Sun, V= inferior planet (Fig 25) J = Superior planet (Fig 26) SA, EA' = direction of mesa 0° from sun .and earth m == sighra amomaly, n = sphuta kendra R = bigger orbit radius = SE in Fig 25 or SJ in fig 26. r = smaller orbit radius = S V in fig 25 or SE in fig 26. K = Sighra karna i.e. distance from earth to planet (true) = EV or EJ. True motion of planets <5 (A'EV) or 6 (A'EJ) But 6 (A'EV) = d (A'EV' - n), and 6 (A'EJ) = d (A'ES-n) For inferior planet, d (A'EV) = 6 (ASV) = sighrocca gati, d n = sphuta kendra gati. For superior planet, <3 (A'ES) = Sighrocca gati (sun is sighrocca for superior planet) (3 n = sphuta kendragati as before Thus in both cases, I 222 Siddhdnta Darpana Sphuta gati = slghra gati - sphuta kendra | gati- (1) | To find <5 n, from figures 1 K cos n - R cos m = r (2) j Differentiating (2), we have - k sin n d n + cos n. <5k + R sin m d m = (3) But k 2 = R 2 + r 2 + 2 R r cos m Differentiating, 2K6 K = - 2Rr sin m <5m - (4) Eliminating (5k between (3) and (4) R r sin m d m x cosn — k sin non- — + R sin m S m= Jv r or K sin n 6 n = R sin m d m (1 - - cos n) k R sin m dm = (K - r cosn) But K-r cos n ,- R cos E R sin m dm x R cos E So on = K sin n But R sin m - k sin n R cos E dm so <5n = — (3) Proof of approximate method. <5 m x R Mandagati phala = - — — dm - mean motion, k - manda karna d m x R Thus mandagati phala = V R 2 + r 2 ±2Rr cosm 2r -1/2 = d m (1 ± — cos m) " neglecting square of -r _ r ~ dm (1 ± — cos m) R .■j True Planets 223 r R cosm <3m 5 <5m± R * R Kotiphala x manda kendragati -&n± . R (4) Approximation of true distance and daily motion can be done by epicyciic or eccentric circles also by successive approximation. That can be seen in commentary on Mahabhaskariya by Prof. K.S. Shukla published by Lucknow University. Geometric explanation of Lalla method can be seen in commentary on Sisyadhivrddida tantra by Smt. Bina chatterjee published by INSA, Delhi-2. Verses 143-150 - Gati phala at four stages for tara grahas- When sighra sphuta gati is more than daily mean motion, then madhyama gati is sub- stracted. When daily mean motion is more, sighra sphuta is substracted from it. When sighra sphuta gati is vakra, it is added in daily mean motion. The result in either of three cases is called first gati phala whose half is taken. When sphuta sighra gati is more, it is added in daily mean motion. When sphuta sighra gati is less or vakra, 1st gati phala is substracted (half only) from madhya gati. This is 1st corrected gati. First gati is forward or reverse. It is multiplied by manda kotiphala and divided by trijya. Half of the result (2nd gati phala manda) is taken. When manda kendra is in 6 rasis starting from karka (90° to 270 °), 2nd gatiphala half is added to 1st gati otherwise substracted from 1st gati. Result will be 2nd gati. Again 2nd gati is multiplied by manda kotiphala and divided by trijya (3438) and full result 224 Siddhdnta Darpang (3rd gatiphala) is used for correcting 2nd gati (addition or substr action). Result is 3rd gati. Third gati is multiplied by slghra koti phala and divided by slghra karna. Result is 4th gati phala, which is used to correct 3rd gati. Then 4th gati is the true gati. If 4th gati phala is more than slghrocca gati then motion is reverse (vakri gati). This method is more correct then surya siddhanta. Verses 151 to 158 - Special methods for true speed- When mandocca of mangala, budha or sard is moving forward, its speed is substracted from 1st and 2nd gati. If it is vakri (in reverse motion), its speed is added. From these corrected, 1st and 2nd gatis, we find 2nd and 3rd gati. New 2nd and 3rd gati are substracted from sphuta slghra gati for 1st and 4th gati. Fourth gati phala of budha is kept in two places. At one place, it is multiplied by manda kotiphala and divided by radius (3438). Result is added to 4th gati phala in second place, when manda kendra is in six rasis starting from karka (90° to 270°). Then sphuta gati of budha will be more correct. If Sighra gati half is vakra, it is added to negative mandagati half or difference is taken from positive half mandagati: Result will be vakra (reverse) gati. Mandagati (2nd or 3rd steps) phalas are added in six rasis starting from karka and substracted otherwise. Many methods of finding true planet from mean planet are coming to mind, but these are not given here (by author), as they are very complicated. . 225 True Planets Daily true motion is used for finding transition time of graha from one rasi to next or in conjunction (yuddha) of planets. For ravi and candra, method is different. Notes : Method for sighragati phala and mandagati phale has already been explained. Reasons of vakra gati will be explained when its starting or ending point are calculated. Verses 159-160 - According to Surya siddhanta tara graha becomes vakri when its 4th sighra kendra has the given values - Mangala 163" Sukra 167° Budha 146* Sani 115" Guru 126° At 4th sighra kendra obtained by substracting these values from 360°, the graha again becomes margi. Notes (1) Derivation of 4th sighra kendra for vakri gati. We assume that heliocentric orbits of earth and Jupiter around sun are both circular and coplanar. « ► u Figure 27 - Explanation of vakri gati and its position 226 Siddhanta Darpana Let S »= Sun, E = Earth, J = Jupiter, u = earth's linear velocity. Sighra karna EJ = k, v= velocity of jupitar. r and R are orbital radii of earth and Jupiter. EE' and JJ' are parpendiculars to EJ so that when relative velocity of Jupiter with respect to earth, i.e. perp to EJ is zero, Jupiter will appear stationary as seen from earth. This means that u cos0 + v cos e - - (1) U - COS £ or — = — (2) v cos w From AESJ, R cos <p + k cos 6 = r - - - (3) r cos <p + k cos e = R (4) t, ,~* , ^w cose rcos0-R ^ From (3) and (4)), = \ ..(5) cos0 R cos <p - r ' COS £ Equating — from (2) and (5) u r cos <f> — R v R cos <p — r ru + Rv cos0 = r=— ^ r v + Ru so that If m is sighra anomaly, then m = 180° - <p Y ru + Rv x So, cos m = - ... (6) r v + Ru This is equivalent to formula given by Bhaskara II ,_ , R cos E . dm spasta gan = sighra gati — i. «. .* ,- » R cos E . <5m .__ spasta gab - 0, if Sighra gab « — — : — - - (7) True Planets 227 i.e. Jupiter appears stationary as seen from earth, if R cosE . 3m sighra gati = ^ u v Angular velocity of earth and Jupiter are - and — so that sun's apparent velocity is also u/r m == Kendra gati = sun's apparent velocity - Jupiters heliocentric velocity. u v_ = 7 " R Substituting this in (7), u Rcos E u v. sighra gati = - = g ( r - p) u E R cosE v_ .-.- (Rcos--l) = — ^— x R Here E = e , R cos E -K = - r cos0 So - x (- r cos 0) = v cos E r or u cos + v cose = which comes to equation (1) Sighra kendra m is obtained from (6) where r = radius of sighra paridhi or antya phala jya, R = 3438', u = mean velocity of sun and v = mean velocity of the planet. (2) Some observations on aghra phala and sighra gatiphala - We have M 2 + E 2 = S ........(1) # Where M 2 = Mandasphuta graha, E 2 = Sighra phala, s=sphuta graha. Differentiating this <5M 2 + <5E 2 = <5S (2) 22 # Siddhdnta Darpat i.e. Mandasphuta gati + sighra gatiphala | = spastagati | (a) Let E 2 be maximum so that (5E = 0, then (5M 2 = <5S This means that when sighra phala is maxi-l mum for sighra kendra 90° or 270° (Sin m maximum) manda sphuta gati is the spasta gati. : ] (b) Planets starts retrograde motion only after; the spasta gati vanishes i.e. <5M 2 + 6E 2 = Taking 6M 2 to be almost a constant, since^ mandagati phala is small, the negative value oft <5E 2 must cancel (5M 2 . (5E 2 becomes negative when sighra kendra is between 90° to 180° or 270° to 360° when value of sine decreases in value. From! 180° to 270° it is negative, hence its net vahiei increases. Thus the planet will have zero velocity" at two points symmetric to 180° (S' towards a and Thus if retrorade motion starts at 180° - it will stop at 180° + = 360° - (180 °-0), where its velocity becomes 0. Keeping earth constant, an inferior planet goes anticlockwise whereas superior planet goes clock- wise which is direction of sun's motion. (c) Values of spastagati at S 7 and C will be by putting R Cos E = R in formula) c * ^ 6- i_ ^ R cos E 6 m Spasta gati = Sighragati — Here, sighra gati = U, dm = U-V K = R + r at S' R - r at c True planets 229 U - R (U - V) Spastagati = ^-^ at S and U - R (U " v ) RV-rU respectively. R + r R- r if R 2 V-2 RV + Rr U - r 2 U > R 2 V - Rr U+rRV _r 2 U i.e. if rR (U-V) > Rr (V-U) i.e. U-V is + ve and and equal to V-U. Thus positive velocity at S' of the planet will be equal to its negative or retrograde velocity at c. Thus velocities direct or retrograde will always be less then S or c at any point between them on either side. Verse 161-164 : Udaya (rising) and asta (setting) of planets is of two types - practical is rough (sthula) and drik siddha is suksma. (accurate) The planets set in west when their sighra kendras cross the following values - Mahgala 332° Sukra 177* Budha 159* Sani 343° Guru 346° For rising in the east, last sighra kendra is Mahgala 28° Sukra 183* Budha 201* Sani 17* Guru 14* For setting in east, sighra kendra of inferior planets are, Budha 310° Sukra 336° 230 Siddhanta Darpana For rising in west, slghra kendra are Budha 50° and Sukra 24° Notes : (1) Rising of a planet means that it is above horizon of earth. But tara graha are visible only during night time, so their rising is only seen at night. Obstruction due to sunrays makes the tara grahas invisible during day. When they are away from sun sufficiently, they can be seen. That is called heliacal rising or drk siddha udaya. (2) Sun's velocity is greater than superior planets, so sun overtakes them so that they set in west and rise in the east. When these planets are situated within particular limits from the sun, they will be invisible in the rays of sun. Thus they will be invisible at conjunction with sun and within particulars limits from position of sun. The total difference from sun depneds not only on difference in longitudes, but also on difference in sara (north south distance.) The limits of invisible distance from sun depends on their distance from sun and relative brilliance. The brilliance also depends on their phase, i.e. part of illuminated disc facing earth. 1 + cos EPS Phase is , ZEPS = sighraphala E2, 1 + cos E2 hence phase = At conjunction E2 = 0, entire planet will be iluminated but we cannot see them, because they True Planets 231 ^vill be immersed in rays of sun. With increase in E 2/ Cos E 2 will decrease and lesser part of disc will be illuminated. Since distance also will decline, luminousity will not be affected, (from S' to a). In path acb, planet gains in illumination and distance also decreases. Thus superior planets appear more and more brilliant when they are retrograding, being most brilliant at c. Spherical radius of jupiter, saturn and mars are in decreasing order, so that they will be visible at angular distances in increasing order. Inverse square law of reduction in brilliance with distance (karna) works but doesn't counter the effect of sizes. Thus sighra kendra of these planets are 14°, 17°, 28°. In udayastadhikara, Kalamsa is slightly less, because distance will be (sighra kendra - sighra phala.) (3) Inferior planets rise heliacally in the east after inferior conjunction and then they are retrograde. They attain gradually the maximum elongation in the east, then direct motion starts. When elongation gradually decreases and after going ahead of sun, they set in east. Thereafter, they heliacally rise in the west. There again, their elongation attains a maximum value, after which they become retrograde. After crossing sun again they gradually set in west and rise in east. (Figure 18 may be seen). When the sighra anomaly of budha and sukra are 50° and 24°, their sighra phala will be 13° and 11°, so that they are the kalamsa i.e. elongation from there mean sun. Then, they rise in west, being near superior conjunction. When their sighra anomalies become 159° and 177°, same sighra phala 2 ^2 Siddhanta Darpana will arise, so that they set heliacally in the west. Then as sighra kendra attains symmetrical values on other side of 180° i.e. (360°-159°) and (360M77*) i.e. 201° and 183 °, slghraphala are same, they rise in the east. Again, when they obtain sighra kendra (360°-50°) and (36°-24°) i.e. 310° and 336\ they set in the east due to same sighra phala or kalamsa. Verse 165 - Moon sets when it is 11° behind sun and rises again when it is 11° ahead of sun. Note : This is not related to rising in east or west. It is visibility near sun, which starts after 11° distance from sun. 12° difference from sun makes 1 tithi (in 360° difference there are 30 tithis 15 in bright half and 15 in dark half). Thus in amavasya, moon is not visible. It is again visible slightly before 2nd day of bright half (12° advance of sun). Thus start of 'duja' in muslim calender is counted from sighting of moon. Verse 166 : To find mean planet knowing the true. Assume the true planet to be the mean; compute the manda and sighra phala and apply them inversely. We have approximation of the mean planet. Treating this as mean planet, again obtain manda and sighra phala and apply them inversely. The process is repeated, till constant values are obtained. Notes : This is method of successive ap- proximation Verses 167-187 - Use of tables for calculation of true planets. Calculation of true planets is very long and difficult process and there are chances of error. :& .'j* True Planets 233 Hence I (author), am giving correct Khandaphalas in a chart for easy calculation (167.) In appendix, there is chart of manda and sighra phala, for parts of to 24 (24 parts of a quadrant of 90° are 3° 45' each). This contains kotiphala of all planets, gatiphala of tara grahas, gatiphala of ravi and candra, sighra of 48 parts (180 °), difference of khandaphala, sighra karna in lipta (minutes of arc), degrees for cakra entry, kranti, sighra kendra for rising and setting etc (170). From the values in the chart, manda kendra bhujaphala in degrees, minutes, seconds etc are separately multiplied by 8, vikala (seconds) etc. are divided by 60, when they become degree, they are added to the degrees. Total degrees are divided by 30 to make rasi. This will be past (gata) phala. (172) For extra degrees, they are multiplied by difference for the khandaphala and divided by 60. Result is added to degrees obtained earlier. Remainder is multiplied by 2 and added to mandaphala khanda. This way, mandaphala of a graha is calculated, which is added or substracted according to rules earlier explained. Manda kendra gati multiplied by difference of khandaphala and divided by 225' (3 °45'), is nianda gati phala between two khandas. (173). * In appendix, parocca khandaphala of mangala, budha, sani also have been given. Khandaphala difference and ucca gati at end of khanda has also been written. From them parocca phala is calculated a nd is added or substracted from manda kendra 4 234 Siddhanta Darpana of mahgala, budha, sani or budha sighrocca, we get sphuta gati corrected for parocca. (174) Sighra khanda table also is prepared for 48 parts (khanda of 180° i.e. 1 part of 3° 45'. Sighra kendra is found by substracting manda sphuta graha from sighrocca. Sighra kendra in 6 rasi's beginning with mesa is caled gata and in 6 rasis beginning with tula it is called gamya. Rasi, degrees etc. of kendra are multiplied by 8 and divided by 60 to get the khanda number (because there are 8 parts of 3° 45' each in 1 rasi of 30") as before. Khanda phala of completed parts is corrected for fraction parts by addition if khanda phala is increasing, or by substraction if it is decreasing. This is sighra phala. (176) If sighra kendra is in first 6 rasis/ khanda phala is added (to manda sphuta graha), or in other six rasis it is substracted. This way madhyama graha is made sphuta by sighra phala half, half mandaphala, full manda phala and full sighra phala. (177) For ravi, candra and marigala, manda paridhi is different for different quadrants. So their mandaphala also has been written for 48 parts of 180° like sighra phala. For value between two khandas, we add fraction of khanda phala dif- ference if khanda phala is increasing. It is sub- stracted when khanda phala is decreasing. Manda phala is never retrograde. (178) For manda phala of mangala, there is no need of calculation between 22nd and 28th khandas. For that interval khanda phala is constant 11*2'47". (174) 235 True Planets Sighra kendra gati is substracted from khanda t>hala and result is divided by 225'. Half of the result is added to madhya gati, if sighra phala is increasing. It is substracted, if sighra phala is decreasing. We get 1st corrected gati. (180) First gati is multiplied by manda phala dfference between two khandas in which 2nd manda kendra lies and divided by 225'. Half of the result is added to first gati, if manda kendra is betwen 90° to 270°, otherwise it is substracted. We get second gati. (181) 2nd gati is multiplied by manda phala difference for 2nd graha and divided by 225. Result is added or substracted from second gati to get third gati sighra. (182) 3rd gati substracted from sighrocca gati gives fourth sighra kendra gati. This is multiplied by khanda phala difference of 3rd graha (manda sphuta) and divided by 225'. Result is added to 3rd gati, if sighra kendra is in 90° to 270', otherwise substracted. We get spasta daily gati. If it is negative, graha is vakri (retrograde). (183) For mangala, budha and sani, vakra mandocca gati is added to 1st and 2nd gati and marg£ mandocca gati is substracted to find the kendragati from mandocca. Mandaphala of this manda kendra is found for second and 3rd gati. sighra.^ (184) If margi (forward) mandocca gati is more than first gati, then first gati is substracted. From remainder second gati will be calculated. Similar method is used for finding 3rd gati. Gatiphala is corrected in reverse manner i.e. substracted for 236 Siddhanta Darpana manda kendra between 90° to 270° and added for other values. (185) This way we get second and third gati of the three planets mangala, budha and sani. (186) If 1st gati of budha and sani is vakra and less than mandocca gati, then it is substracted to get second gati. If vakra gati is more, mandocca gati is substracted from it but mandagati phala is added or substracted in opposite order. (187) Verse 188 - If in chart of khanda phala, some khanda phala is missing or unclear, then its khanda number is multiplied by 225' and for kendra of that kala, we find bhuja and bhuja koti. Verses 189-191 - Difference from sphuta surya in degrees is given at which a graha sets due to sun rays Vakri Sukra 7°, Sukra (margi) 9° Guru 10°, Chandra 11° Budha 12°, Sani 14°, Mangala 16° These values in degree are multiplied by 1800 and divided by rising time of the rasi in which sayana sun is situated. This will be ksetramsa. If it is in west, then 6 rasi is added to the result. Then ksetransa is substracted from (sayana sun + 6 rasi.) When mangala, guru and sani are less than ravi by at least the ksehansa, they rise in east before sunrise. When they are ahead of sun by ksetramsa, they set in west after sun. (Thus they are visible only in night). When vakri budha and sukra are behind ravi by this ksetransa, they set in east and when ahead of ravi, they rise in west. True Planets 237 (Just before sun rise, since sun is coming upon horizon, they go down being vakrl. During night, they are visible when sufficiently away). Similarly, they rise in west just after sun set when vakri). Notes : (1) Rising times of rasis is explained in Triprasnadhikara. Briefly, rasis rise in different time because it is oblique with equator (23-1/2)°). At places farther from equator, obliquity rises and difference in rising time of rasis increases. This calculation is done for sayana surya, because surya goes on equator when sayana surya is at 0° or 180°. Roughly the planets are assumed in same plane as sun, as their inclinations to ecliptic are very small. So rising time for their difference along ecliptic will be same as rising time of sayana sun for that rasi. Since rising time is given for 1 rasi of 1800 kala in asu, equivalent difference on ecliptic is given by multiplying given degrees (kalamsa) by 1800 and divided by rising time of rasi. This is almost same as kalamsa, being its projection on ecliptic. (2) When planets are behind sun, they rise before sun in east, if difference is more than kalamsa. Being behind, earth horizon in east meets them after wards. Vakri budha and sukra have already been explained. Verses 192-193 : Finding time of udaya or asta From the kalamsa given we can calculate the time in days since when graha has set or risen (heliacally). If their difference with sun is more than Kalamsa, the planet has already risen or set. If it is less than kalamsa, the time to reach kalamsa 238 Siddhanta Darpana can be calculated, which will be days after which planet will rise or set. (Difference of planet and sun - kalamsa) is divided by difference in speeds of sun and the planet. The no. of days will be found since when planet is rising (or setting) or after which it will rise again. Verse 194 : Start and end time of rising and setting of planets should be written in the practical calender, because it is very difficult to find it by drk karma. * Verse 195 : In appendix, khandaphala and their differences are given. Similarly differences of gati phala, and karna (in kala) also should be calculated and written. Sighrakarna, gati and sphuta positions etc will be found by values given for places just before the given position. Difference of phala is to be added or substracted when the value (phala) is increasing or decreasing. Verse 196 : Frequency for finding true positions - Sun and moon should be made sphuta every day at sunrise time. At end of a paksha, all graha should be made sphuta. Budha should be made sphuta in middle of paksa also (i.e. every week). When a planet becomes margi from vakri or vice versa, or changing from one rasi, naksatra to another, or start of rising time or setting should be calculated more accurately by method of successive approximations. Verse 197 - There are 200 kala (minutes of arc) in a quarter of a naksatra, 800 kala in a naksatra and 1800 kalas in a rasi. To find the days since when the graha is in a particular rasi, naksatra or True Planets 239 quarter of a naksatra, we take the difference of rasi etc of graha and the rasi etc of the beginnig of rasi, naksatra or its quarter. The difference is divided by sphuta gati kala. Result will be days etc since when the graha had entered that rasi etc. When graha is less than rasi of naksatra etc, the reverse difference will be divided by sphuta gati. Result time in days etc. will give the period after which graha will enter that naksatra etc. When graha is vakri, opposite process will be done. Notes : (1) Ecliptic of 360° has been divided into 12 rasis and 27 naksatra of equal interval. Hence 1 rasi = 30° = 1800' Kala 1 naksatra = 13° 20' = 800' kala 1 naksatra quarter (1/4 or pada) ■ 3° 20' = 200' kala (2) Rasi's starting from 0° of ecliptic are (1) mesa (2) vrsa (3) mithuna (4) karka (5) sirnha (6) ka'nya (7) tula (8) vrstika (9) dhanu (10) makara (11) kumbha and (12) mina Naksatras starting from 0° of ecliptic are (1) asvini (2) bharani (3) krttika (4) rohini (5) mrgasira (6) ardra (7) punarvasu (8) pusya (9) aslesa (10) magna (11) purva phalguni (12) uttara phalguni (13) hasta (14) citra (15) svat! (16) visakha (17) anuradha (18) jyestha (19) mula (20) purva asadha (21) uttara asadha (22) sravana (23) dhanistha (24) satabhis (25) purva bhadrapada (26) uttara bhadrapada (27) revati. (3) Within a rasi or naksatra a graha can be assumed to have the same true motion hence the formula uses the relation- 240 Siddhdnta Darpana 7*. 1 Distance in kala = days X speed per day in kala. (4) Candra moves faster and position of candra and sun are to be known accurately for start of day, tithi etc. Hence they are to be calculated each day. Other planets are not so important so they can be calculated each paksa (fortnight). Budha moves faster, hence its calculation should be done twice in a fortnight. (5) For change of vakri or margl gati or rising or setting times, the speeds change within a day also. Hence calculation needs to be made accurate by method of successive approximation. Verse 198 : Dainika spasta gati of a graha can be found roughly by taking difference of spasta graha at beginning and end of the paksa (fortnight) and dividing it by number of days in it (round figure of 14 or 15 when days are counted from sunrise to sunrise) Difference between spasta graha on two successive days at sunrise is more accurate dainika gati which is useful for calculation. Both differ very little, so very little error is made if we take average daily speed for a paksa. If the two are different, then method of successive approxima- tion is used. Verse 199 - Fourth slghra kendra is calculated at the end of every paksa. As already stated, slghra kendra of graha for which it becomes margi or vakri, its rising and setting has been given ir^ appendix. To find the position of slghra kendra at any time between paksa ends, divide the difference beetween values at end with days of paksa and add them proporitionately for the time passed. ■1 rJ. y V"- True Planets 241 Veise 200 : 21,600 kala divided by 30, 27, 12, 27 and 60 gives measures of tithi, naksatra, rasi, yoga and karana, i.e. 720, 800, 1800, 800 and 360 kalas. Notes : (1) Tithi, naksatra, yoga, karana and vara are five parts of a calender - hence it is called pancanga. Vara is successive counting for days starting from sunrise, hence no calculation is needed. (2) Definitions - 'rasi' is 30° part of the ecliptic where planets move. Rasi of a planet means its completed rasis from 0° of ecliptic as well as degrees, minutes, seconds, lapsed in the current rasi. Though it is not part of pancanga, it is used to calculate all other parts. Naksatra is found by dividing ecliptic into 27 equal divisions of 13°20' each (total 360° = 27 X 13° 20') Each part is naksatra. 'Naksatra' mentioned in pancanga means the naksatra which is occupied by moon at a particular time. i.e. the time when moon goes one circle more than sun. It is measured usually from the time when sun are moon are together, i.e. difference between their rasi is 0°. That is start of first tithi called amavasya, i.e. when sun and moon live (vasa) together (ama = amity = closeness) Month can also b e counted from time when sun and moon are in opposition (i.e. 180° away) Then full moon is seen, so that is end of purnima tithi. The two systems °f lunar month are called amanta (ending with a navasya) or purnanta (ending with purnima). 242 Siddhanta Darpana Tithis are not counted serially from one to 30 in lunar month. They are counted from 1 in each half (Sukla = bright and krsna = dark) In sukla paksa last tithi is written 15 and in krsna paksa it is written 30 (denotin end of month). Since 360* difference between moon and sun causes 30 tithis, 1 tithis is result of 12° difference. Thus difference of 0° to 12° is 1st tithi in sukla paksa after amavasya, 12° to 24°/ 2nd tithi etc. upto 180° the paksa will be sukla paksa with 15 tithis. Between 180' to 360* difference it will be krsna paksa with 15 tithis. Thus the number of complted tithis Moon - sun " 12° Fraction will give the part elapsed in the current tithi which is next after completed tithi. When the quotient is more than 15, than 15 is substracted to know tithi of krsna paksa. Karana is half part of tithi, caused by 6" difference between moon and sun. Thus completed karana since amavasya end Moon — sun These are not counted from 1 to 60 in a month, but there is rotation of 7 karanas like 7 week days, 8 times in a month and 4 remaining karanas are given separate names fixed at both ends of a month. This is explained later in detail. Karana and tithi both indicate the phase of moon, i.e. the fraction of its disc which is illuminated. Naksatra and rasi of moon (or any True Planets 243 other planet) can also be physically seen. But yoga is not a physical quantity. It is only a mathematical function given by sum of rasi etc of moon and sun (for tithi and karana, their difference had been taken). However, one full revolution of moon + sun is not divided into 30 parts like a tithi, but in 27 parts only like a naksatra. Thus for each increase in sum of moon and sun by 13 °20' one yoga passes. Thus number of completed yogas counted from time when sun of moon + sun was 360° or a is Moon + sun 13° 20' List of yoga is given later. (3) In a full circle there are 21,600 lipta or kala. Hence measure of naksatra etc is found by their total number in circle by which 21,600 is divided. Verse 201-202 - Calculation of tithi - Time lapsed (gata kala) and remaining time (gamya kala) of the current tithi is found by dividing difference of moon and sun in kala by 720 kalas. Remainder is converted to vikala (on multiplication by 60). This will give gata kala. Dainika gati of ravi and candra is found by difference of current day and next day's position. Gata or gamya tithi is divided by difference «of dainika gati of moon and sun. This will give value m danda etc. (when gata tithi was in vikala). This is rough approximation, sufficient for normal work. ^ this we have used dainika gati for 1 savana dl na in stead of gati in 1 tithi. If further accuracy 244 Siddhdnta Darpana is needed, we find gati of a tithi from dainika gati and ravi, candra are further corrected. Verse 203 - Lapsed or remaining time in rasi or naksatra — Sphuta kendra is converted to kalas and divided by 800. Quotient will be number of past (gata) naksatras counted from asvini. By adding 1, we get the number of current naksatra. Remainder is the lapsed part (in kala) of the current naksatra. Substracting this from 800' we get remaining part. It is multiplied by 60 to make vikala and divided by dainika gati (in kala). This will give lapsed (or remaining) time of naksatra in danda etc. Sphuta candra converted to kala and divided by 1800 kala in a rasi gives number of completed rasis. By adding 1 to quotient we get the number of current rasi, counted from mesa. Remainder will be lapsed part (in kala) of the current rasi). It is substracted from 1800' to give remaining (gamya) part. Gata or gamya part is converted to vikala by multiplying with 60 arid dividing by spasta dainika gati of candra. We get gata or gamya kala of the current rasi in danda etc. Note - Gata or gamya part (in kala) — x = 60 x vikala. kala Dainika gati = Difference in position in 1 day = -r— gata part xkala Hence *j — = — x day = 60x danda gab Kala/ day J Hence n is converted to vikala before division by gati. Verse 204 — Calculation of yoga Add the rasi of sphuta candra and surya. If it is more then 12 rasi's, substract 12 rasi from the sum. It is converted to kala and divided by 800 = True Planets 245 nC . of kala in a yoga. Quotient will be number of completed yoga counted from viskumbha. Add 1 to it, we get number of current yoga. Remainder gives part of yoga lapsed in kala. By substracting it from 800', we get remaining part of current yoga. Gata or gamya kala is multiplied by 60 to make it vikala and divided by sum of dainika gati of sun and moon. We get gata (or gamya) time in danda etc. Note : List of yogas .— (1) viskumbha (2) priti (3) ayusmana (4) saubhagya (5) sobhana (6) atiganda (7) sukarma (8) dhrti (9) sula (10) ganda (11) vrddhi (12) dhruva (13) vyaghata (14) harsana (15) vajra (16) siddhi (17) vyatfpata (18) variyana (19) parigha (20) siva (21) siddha (22) sadhya (23) subha (24) sukla (25) brahma (26) aindra (27) vaidhrti Verse 205 - Calculation of karana ■ Add 360 kala to spasta ravi in kala. Deduct the sum from sphuta candra in kala. Divide the difference 360 i.e. no. of kala in a karana. Quotient is divided by 7. Remainder is number of completed karana. By adding, we get the current karana. Karana starts from second half of sukla 1st day with 'Bava'. After end of seventh karana, again first karana 'bava' starts. In 30 tithis of candramasa, there are 60 karanas. 7 Karanas are repeated 8 times. Remaining 4 karanas are fixed (sthira) wlfich are sakuni, naga, catuspada and kinstughna. Note : (1) Moving karanas start after 1st half °f 1st tithi (sukra 1st tithi) has already passed. Hence 360 kala is added to ravi so that in difference from moon, 1 karana is deducted. 246 Siddhanta Darpana (2) Seven moving karanas (chala karana) are - (1) bava, (2) balava (3) kaustubha (4) taitila (5) gara (6) vanija (7) visti or bhadra. Last karana is considered inauspicious for good work. Similarly Sunday was not supposed a day for doing work out of seven week days. (2) Sthira karanas sakuni, naga, catuspada and kinstughna start from krsna 14th second half, 30th (15th krsna both halves) and sukla 1st tithi. (3) In vedanga jyotisa, 11 karana or half days were deducted from solar half year (equinox to next eqinnox in opposite direction) to make it equal to lunar month. 371 tithis in a solar year are divisible by 7, though 365 days are not divisible, hence fraction of weeks remain. Similarly in half year, karanas (half tithis equal to 371) are divisible by 7. Out of 11 karanas last 4 are fixed, as in a month also 4 remain after 7X8 cycles of 56 karanas. Verse 206 - If at time of sunrise, the total gata and gantya kala of tithi (720) is more than the difference in dainika gati of candra and ravi (i.e. difference is less then 720' per day), then tithi is long (tithi vrddhi) i.e. more than 60 dandas. Tithi vikala 720 x 60 divided by difference of candra and ravi gati, we get duration of tithi in danda etc. If it is more than 60 danda then there is tithi vrddhi, otherwise tithi ksaya occurs. Verse 207-209 - Extra and ommitted candra months- When in a candra masa, there is surya sankranti (i.e. surya goes from one rasi to another), then it is called suddha candra masa (i.e. normal month). When there is no surya sankranti (i.e. surya remains in same rasi), it is called extra month True Planets 247 (mala or adhika masa.) Next amanta month is called normal candra masa. When there are two sankrantis of surya in a candra masa, it is called ksaya masa (lost month) - i.e. next, candra masa is not counted. Before and after ksaya masa, within 4 months there are one mala masa each i.e. two mala masa in that year. First mala masa is called sansarpa, ksayamasa is called amhaspati and later malamasa is called mala. Both mala and ksaya masa are prohibited for any auspicious work. (207) In veda and smrti, the works which are prescribed, monthly and annual sraddha can be done in sansarpa or amhaspati, but not in the later malamasa. Malamasa is counted as a month for annual sraddha of a dead man, when it comes within start and completion of a month. New work is not started in a malamasa, but work started earlier can be continued. The following works can be done in a malamasa- Bath during eclipse, charity, observing rare yogas (auspicious times), sudden works, promised work, coronation, santi, pusti karma, functions related with child birth, sraddha etc. (208) A ksaya masa is repeated after 141, 122 or 19 years. In current year (1869 when book was written) mandocca of sun was in mithuna, hence in 9 months from phalguna, a mala masa is probable. 3rd months after karttika may be ksaya masa, Magna month may be ksaya or adhika. Notes : (1) A lunar synodic month is approximately 29.5 days long, where as surya remains in a rasi of 30° for 30.4 days. Thus lunar month is completed earlier and after about 30 24# Siddhdnta Darpana months extra days in solar month will amount one month and sun will not cross to next rasi. Example of mala masa is explained below - a p y d y Sravana Bhadra Asvina Karttika Margslrsa Pausa A B C D E F G ABCD are kranti of sun. Signs on upper part denote start of a lunar month. In Bhadra there is no sankranti so it is a mala masa. (2) Frequency of malamasa - There are 1593336 malamasa is 51840000 solar months of a yuga i.e. 66389 adhikamasa in 2160000 solar months. 66389 __LJ__LJ_J_1 1 2160000 " 32 ' +1 ' +1 ' +8 ' +1 ' +1 ' +5 n L 1 1 2 13 15 25 Convergents are -,-,-,— , — / ^ I 1 J 2 — and — are on either side of the true figure. Hence adding numerator and denominator both, we get a better approximation. Thus 3/98 is ratio of adhika masa i.e. 3 adhika masa in 98 months (solar). (3) Adhika masa and year — There are 1,593,300,00 adhika masa in a kalpa of 4,320,000,000 years i.e. 5311 adhika masa in 14400 years 14400 1 1 1 1 1 11 i i 5311 1+ 2+ 2+ 6+ 1+ 1+ 7+ 8+ 2+ Successive Convergents are True Planets 249 2 1 3 19 122 141 T ' 3 ' 8 ' 7 ' 55 ' 62 Thus there are approximately 7 adhika masa in 19 solar years which was used in vedanga jyotisa (Rk veda). This was known in Romaka siddhanta and was called Metonic cycle in Greece. Next approximations also indicate possibility of ksaya masa in 19, 122, 141 years. Verse 210 - Thus the rough pancanga with its components like tithi and naksatra is completed which may be accepted by the learned and they may perform every year the daily, occasional and conditional functions, fasting days, sraddha, fes- tivals etc. according to this pancanga. This may do good of world as it is according to jyotisa samhita and well thought of. Verses 211-212 - For daily auspicious functions I am preparing this pancanga with positions of sun and other planets. While doing the work I pray to lord Jagannatha who is on nilacala shining like black soot (for eyes). Thus the fifth chapter describing true planets with their khanda phalas is over in siddhanta darpana written for education of children and calculation as per observation by Sri Candrasekhara born in a famous royal family of Orissa, Chapter - 6 CORRECTIONS TO MOON Scope - Accurate panjika and further correc- tion to motion of Moon General Introduction (1) Equation for elliptical orbit round earth. Eccentricity of moon is 0.0548442 = e So e 2 = 00.0030079, e 3 = .00016496 e* = .00000905 Higher powers e 5 etc are very small and can be neglected. Thus 6 measured from mandanica or perigee is given in terms of position m of mean planet as * = m + (2e-±e 3 + ^) sinm 96 5 2 11 4 „ e 6 + i 7 "24* + 17 I^)sin2m 13 3 43 5 { 12 6 ~ 64* ^ Sin 3m 103 4 451 6 1097 e 5 ( 16* 480* > Sm4m + ~lio- sin 5m ... = m + (0.1096884 - 0.00004124) Sin m + (0.0037599 — 0.00000415) sin 2 m + 0.0001787 sin 3 m + 0.0000097 sin 4 m = m + 0.10964716 sin m + 0.00375575 sin2 m + 0.0001787 sin 3 m + 0.0000097 sin 4 m Corrections to Moon 251 The sine ratios in radians are converted to 1 i_ 180 ° kala fer degree) by multiplying with ~ ir x 60 = 3437.75 kala or 206265 vikala. Then $ = m + 376'56". 4 sin rn + 12'54". 7 sin 2m + 36". 9 sin 3 m + 2".0 sin 4 m Here m has been calculated from nica or prigee. If it is calculated from apogee or mandocca, then 6 = m - 376'56." 4 sin m + 12'54"7 sin 2 m - 36".9 sin 3 m + 2". sin 4 m Here m on right side is manda kendra - i.e. distance of madhya graha from mandocca of moon. Remaining terms are mandaphala. When 6 = 90°, sin m = 1 and sin 2 m = Then highest mandaphala depends only on its first term 377' approximately or 6*17'. But our astronomers have taken highest mandaphala about 5° only (radius of mandaparidhi of 32°). However, on new moon or full moon day, when moon is 90° away from mandocca, then it is V2(Y ahead of its calculated position. When moon is 270" ahead of mandocca or 90* from nica then it is 1° 20* behind its calculated position. Thus in both situations mandaphala correction is 6'16'56".4-r20' = 4°56'.4 (correction is-ve for m = 90° and positive f or m = 270°). Thus maximum mandaphala is about*5° only as observed. However, in middle of a paksa i.e. on 8th day, if this mandaphala correction for manda kendra 90° is taken as 5°, then observed moon is 3° behind calculated moon or 8* behind mean 252 Siddhanta Darpana moon. Thus cauculations in our siddhanta were true for purnima or amavasya when eclipse is to be calculated. One reason for such neglect is that accuracy is needed only for eclipse, other reason is that observations were done ony on purnima or amavasya days or more accurately at time of eclipse. This is still followed by muslims and even now eclipses are studied for more accurate observation. (2) Deviations in moon position due to effect of sun - Effect of sun is three types (a) Attraction component of sun on moon in direction of earth moon radius, elongates the orbit in the direction of sun and away from it. It changes eccentricity of orbit and is called evection term. Since it changes eccentricity of orbit, called 'cyyti' it was called 'cyuti' sanskara by Sri Venkatesa Bapuji Ketakara in his Jyotirganita. Since it changes angle from mandocca (or Tunga = top), it has been called Tungantara' sanskara in siddhanta darpana. (b) Component of sun's attraction on moon in direction of moon's motion advances it towards sun, which is maximum in middle of a paksa and nil at its ends. This varied speed, hence it was called variation. Its frequency is in 1 paksa, hence it is called paksika sanskara in siddhanta darpana. Sri Ketakara called it tithi sanskara because it depends on tithi of the paksa. (c) Due to difference of sun's distance from earth or moon depnding on its direction from earth, its attraction force on moon varies in a period of 1 year. This is called digamsa sanskara as it amounts to 1/10 of sun's equation. This is also called varsika sanskara because its period of variation is one year. i ■n Corrections to Moon 253 Figure (1) (a), shows force of attraction G due s* Fig 1 (a) Fig 1 (b) Fig 1 (c) Figure 1 - Effect of sun's attraction on moon's orbit to sun. In positions A and B which are near to sun compared to earth, extra attraction on moon is in direction of sun. In position C and D of moon, away from sun, the difference in force compared to earth is away from sun. Force of attraction G has two components, its component R is reducing the pull of earth on moon acting in opposite direction. Thus distance of moon increases from earth. This increase is maximum for positions Mi and M3 and nil for positions M2 and M4- Thus in Fig 1 (b), when major axis is in direction of sun, the axis will become longer and its eccentricity will increase. In fig 1 (c), the distance perpendicular to ma Jor axis in sun's direction will increease, due to which moon orbit will become round. Then e ccentricity will decrease. Thus correction in man- 254 Siddhanta Darpana daphak due to eccentricity will increase for fig (1) (b) and decrease in position of figure (c). and ^T V ° ne ?u T iS maximum for position M« and mcreases the speed in middle of krsna paksT of Z C T *?* beC ° mes zero at M > « *e force of attraction ,s totally in direction of EM,, aiS other component is zero. It increases in value from m! o M T 3gain d6dineS to 2er <? at M» ^om M, to M 2 it is against the direction of motion. It is in direction of motion between M,, M, (decreasing) and again increasing upto M< b* against the motion. A««5? Correction by different authorities - of m S ™ ^° dem astronom y, Principal terms or moon s motion are - = m ■+ (377'19".06 sin m + 12'57".ll sin 2 m correct/ 9 Sln 3 m + 2 "° ^ *»> -ndaphala + 1-16-26" sin [(2 (M-S) - m] - - Evection or Tungantara + 39'30" sin 2 (M-S) - - variation or paksika + ll'lO" sin (manda kendra of sun) - - - Annual or digamsa onlv 'Si Carly / S u°f ° merS ° f ^ cognised ™LK, mand ^ ya correction, (equation of centre), but instead of its value to be 377' sin m they took its value 301' sin m, by includtag effeS 301' sin m + 76' [(sin m + sin {(2 (M-S) - m ]]+ Corrections to Moon 255 = 301' sin m + 152' sin (M-S). Cos (S-a) Here M = mean Moon, S = True sun, a = moon's perigee from which angle m has been measured. Thus m - (M-S) = S - (M - m) =s-a in cos term above. Value of 1st correction to moon was the following according to different authors Aryabhatiya 300'15" Sin m Khanda khadyaka 296' sin m Uttara khanda khadyaka 301'.7 sin m Brahma sphuta siddhanta 293'31" sin m Greek value 300'15' sin m Siddhanta darpanaNj00'49".5 sin m Surya siddhanta 302'23".66 sin m Bhaskara H, 301'46".8 sin m (a) Second correction term by Manjula (932 AD) In Laghumanasa, 1st mandaphala correction of moon has been given as 488' sinm — degrees 97' + — cos m where m is mandakendra measured from apogee. Thus maximum value of mandaphala is for m = 90°, 488 ^ degrees = 301'50" Second correction has been given by 8° 8' cos (S-U) (True moon - 11) x 8° 8' sin (M-S) Siddhdnta Darpana where S,M,U are true sun, true moon and mandocca of moon. For simplicity, daily motion of 790 °35" of moon is taken as true motion, then this becomes 8 B 8' x 8-8' x V 11' cos (S-U) sin (M-S) = 144-26' cos (S-U) sin (M-S) - converted to minutes, 2nd correction (l) Thus Manjula's correction is sum of two correction - (i) 76' sin (M-U) - part of the mandaphala (ii) 144-26" cos (S-U) sin (M-S) - - - evection term which was not mentioned by previous astronomers. Plotemy had given maximum value of 2nd correction as 159' but didn't give any formula (150 Astronomer Yallaya gives credit of this dis- covery of these corrections to Vatesvara (904 AD) but this has not been found in Vatesvara siddhanta the available book. ' This appears in exactly the same form in karana - kamala-martanda of Dasabala (1058) Subsequently it occurs^ in equivalent forms in siddhanta sekhara of Sripati (1039 A.D), Tantra Sangraha of Nila Kantha (1500 A.D.), uparaga kriya krama of Narayana (1563 A.D.) Karanottama of Acyuta (1621 AD) and lastiy in siddhanta darpana of Candrasekhara (1869). " Equation (1) of Manjula is correct but constant is 8, less. Snpati's second correction amounts to the following correction term. Corrections to Moon 257 160' cos m sin (mandaphala) x 1 - cos (mandaphala) Mandakarna - R where R = radius 3438' This is same as Manjula's equation except that the constant is now. 8' more, instead of 8' less earlier. (b) Bhaskara II - Bhaskara II wrote a separate work called 'Bijopanaya' about corrections needed in true planets. Stanza 8 of the work starts with statement - I have seen maximum difference between calculated and observed positions to be ± 112' When moon is one quadrant ahead of man- docca and sun is half aquadrant ahead of moon, observed moon is 112' behind calculated moon i.e. negative error. When moon is 3 quadrants ahead of apogee and sun at half a quadrant behind her, the maximum positive discrpancy of + 112' is seen When eclipses of sun and moon take place and moon is at apogee and perigee, there is no error or bija. When eclipses take place at end of odd quadrants from apogee, error is negative equal to TA r, . and sun is When moon is at the apogee, and :.-;_ la ahead or behind by half a quadrant, descrepancy Same discrepancy is seen, when moon is at P e ngee and sun half quarter ahead or behind His first equation of mandaphala was correct 258 Siddhdnta Uarpana = - 301'46" sin m. . But after Bijopanaya he gave the equation - 379'46".8 sin m + 34'sin2 (M-S) where m is manda kendra, M and S are true moon and sun. His new equation totally missed the evection term, and it became more incorrect at eclipses; though his observations about error were correct. (c) Correction by Candrasekhara - His first equation of apsis (mandaphala) is (31° 30' - 30" cos m) 3438 sin m 360° = - 300 / 49 ,, .5 sin m + 4'46".5 sin m cos m = - 300 / 49".5 sin m + 2'23".25 sin 2 m Though he has attempted to correct the second order of small quantities, his constant is too small (l/5th of the correct value). (2) Tungantara correction is of the form 1W x 3438 sin (a - S - 90) 3438 sin (M - S) 3438 X 3438 (where a is apogee of moon) Moon's true deaily motion x ' ' Daily mean motion = - 16C cos (Sot) sin (D-0) Moon's apparent deaily motion Daily mean motion (3) Paksika equation or variation in Daily mean . . 3438' sin 2 (M-S) _,„_„ . -/wcv motion is ~> — — - = 38'12" sm 2 (M-S) 90 ,3 Corrections to Moon 259 Here the constant is less by 1'18" from modern value. (4) Digansa sanskara for annual variation is 1 12 x 3438 . „ ■*■ — x zm sin S m ~ 10 360 (Sm = manda kendra of sun) = ± 11'27" sin Sm Modern value of the constant is ll'lO". Tycho found it to be 4'30". Horrocks' (1639)) found it 11'51". He has indicated in the text that new equations were to correct the discrepancies ob- served by Bhaskara II, in which he was brilliantly successful. (4) Modern charts for calculating moon's position - Constants of moon's motion at 1900 AD, 0.0 day epoch is Mean longitude L = 294°. 56984 + (1336 r) 307.8905722 T + 0.00918333 T 2 + 0.00000188 T 3 Mean anomaly M = 229°. 97832 + (1325 r) 198°51'23".5T + 44".31T 2 + 0".0518T 3 Mean longitude of node V = 259 Q 12'35".ll - 6962911". 23 T + 7". 48T 2 + 0.008T 3 For perturbations the constants are given by Hansen as- Ao = 69.80458 + (1148r) 55.37787761T + 0\00881085T 2 + 0\0000011374958T 3 Bo = 352.81434 + (2473r) 254" 23441630T + °. 000420645 T 2 + 0* .00000301393 T 3 Co = 204°. 85020 + (99r) 359*.051667T + 0.0001988055T 3 260 Siddhdnta Darpana 1 D = 190 '.45443 + (1048r) 56 '.32271091 T + I 0°. 007903044 T 2 + 0°. 00001 1374958T 3 E = 354°. 45312 + (2373r) 255°. 17924960 T + 0° 004405255 T 2 + 0°. 00000301393 T 3 F = 341°. 85083 + (1131r) 172°. 20183595 T + 0°. 00430092 T 2 + .000003347264 T 3 Components of perturbation effect are - A = 4467" Sin Ao = 1.24083° Sin Ao B = 0.59583 Sin Bo C = 658" Sin Co = 0.18277 Sin Co D = 0.55 Sin Do Total effect of perturbation = G = A+B+C+D+E Perturbation in latitude is F = 0.1453 sin Fo From the value of these constants equation of centre and latitude is calculated. (5) Indian Charts - In India also many charts were prepared from time to time. Makaranda sararu was most famous. Candrasekhara has referred to tables of Kochanna of Andhra pradesh. Then in south India, specially in Kerala, vakya karana are very famous. Original Vakya karana was written for moon—called candravakyani by Vararuci, reputed to be in time of king Vikramaditya at start of Vikrama eera. Then Vakya karana was prepared in 13th century. Its writer is not known, but Sundararaja commentary is available. These books calculated the days from kaliyuga beginning. The moiton was calculated fof a convenient lump of days. For remaining number of days, the true position was calculated at about Corrections to Moon 261 200-300 positions. These were indicated by (vakya' for each of position to be read in Katapayadi notation. This method could give correct position upto minute for 24 hour intervals. Madhava of sangamagrama in 1350 AD, prepared 'Sphuta candrapti' to calculate true moon upto seconds of arc at 9 periods in a day. His method was to calculate position of moon at equal intervals of 24 hours from its mandocca position. Moon reaches from mandocca to mandocca in about 248 days, so 248 vakyas are used. (6) Making of a calender - One of the main aims of astronomy is to find suitable measurement of time. A time scale to indicate past time since an epoch is a calender. Intervals of time which can be measured is one type of kala and its measurement is called 'kalana' Thus 'calculate' means to count or to measure. In Arab, they were called 'kalamma' Work of 'kalana' is called chronology or calender. The flux of time is apparently without beginning or end, but it is cut up periodically by several natural phenomena- (i) by ever recurring alteration of day light and night (ii) by the recurrance of moon's phases (iii) by the recurrance of seasons These have been used to define natural divisions of time- Day - time of alteration of day and night Month - Complete cycle of moon's changes °f phase - 262 Siddhdnta Darpana New moon to new moon (amanta month) or full moon to full moon (purnimanta) months. Year - Coming back of a season again and its smaller subdivision season. Standards for day - Day for purpose of regular works was counted from sun rise to sunrise in India and from sunset to sunset in west asia (Babylonians and Jews). West Asia was called 'Asura' area and hence they were called nisacara (moving in night) because their day started from night time. Sunrise and sunset are convenient to see and day light only gives opportunity for doing works. Sunrise time varies according to position of sun in south or north hemisphere of sun. Variation of day length is more in places away from equator, being nil at equator. Hence for calculation purposes day was counted from midnight to midnight. Even midnight to midnight day varies, be- cause during this time earth makes one rotation arounds its axis with respect to stars and has to move further to catch up with movement among stars. This second component varies with distance of sun which varies in an elliptical orbit. Thus revolution of earth with respect to stars is taken as a better standard called sidereal day. An average of solar day (midnight to midnight) is used and called mean solar day. 1 1 , . , 365 - mean solar days = 366 - sidereal days 1 1 hour = — of mean solar day. i ■2a ■§ 263 Corrections to Moon Rotaton of earth = 23 h 56 m 4.100s mean solar time Sidereal day = 23h 56m 4.091s mean solar time Mean solar day = 24 h 3m 56.555s sidereal time Slight variation in rotation period of ear* and sidereal day is due to obliquity of earth, rotation being counted in the ecliptic plane. Even earth's rotation period is not constant but fluctuates regularly and irregularly by amounts of the order of 10* seconds. Regular slowing down of rotation period is 14 seconds per century due to tidal friction caused by difference of attraction force on sea water in different parts of earth. It is mainly by moon and l/4th by sun. Irregular variation is due to- force exerted by wind movements or unequal rate of atmospheric rotation and sea currents, both of which are caused by heat of sun. Month - Period from new moon to new moon varies from 29.246 to 29.817 days due to eccentricity of moon's orbit and other causes like effect of sun. Period of mean lunation is given by 29.5305882-0.0000002 T days where T = no of centuries after 1900 AD. It may be noted that this is not the period of rotation of moon round earth. This is extra one round ahead of sun. When moon and sun are together, it is anuivasya (living together). Moon with its faster motion goes ahead in about 15 days by 180' when it is purnima (or full moon). After 29.5 days it is again with sun. This rotation is with 264 Siddhanta Darpana speed (moon-sun) and slower than moon's rotation in 27.3 days only. Year and seasons - 1 year is one rotation of sun with respect to stars - it is called sidereal year. Seasons change according to position of sun with respect to earth in north south direction. It is perpendicular to equator twice in one year, while coming from south to north it is called vernal equinox and in opposite direction it is autumnal equinox. Equinox means equal day and night (nakta in sanskrta = night) If axis of earth is fixed, tropical and solar years will be same. But it rotates in reverse direction in a conical manner, thus equinox points rotates west ward making a rotation in about 25000 years. Due to this precession of equinoxes occurs. Tropical year = Sidereal year - speed of precession per year (crossing time by sun) Present values are Tropical year == 365.24219879 - 0.614 (t- 1900)xl0" 7 days, where t = Gregorian year Thus it is 365.2421955 days = 365d 5h 48 m 45.7 sec. Sideral year is 365.256362 days. Only tropical year corresponds to the seasons In addition to two equinoxes, we can take the points of longest day (in north hemisphere) where sun is northern most from equator i.e. summer solstice or the southern most position called winter solstice. As the day is counted from midnight i.e. lowest position of sun in east west circle, year can Corrections to Moon 265 be counted from southern most winter solstice (which is lowest for northern hemesphere). This is like a grand day hence one tropical year is called a 'divya dina' (divine day). Since the grand day starts with winter solstice from vedic days, the first day 'christmas' is called 'bada (grand) dina'. Actually it is start of grand day. That month called margasirsa has longest nights hence it is called Krsna masa (or black month). Thus Krsna has compared himself with margstrsa month in gita. This has become 'Christmas' (Krsna masa). 15 days before start of margasirsa masa will be beginning of great usa (Twilight before sun rise), hence it is called 'bada osa' in local languages (like in Orissa) Problems in calender making - Civil calender for use in human life has following difficulties (a) Civil year and the month must have an integral numbers of days - perferably equal (b) Starting day of the year, and of the month should be suitably defined. The dates must correspond to seasons. (c) For the purpose of continuous dating, an era should be used and it should be properly defined. (d) The civil day, as distinguished from the astronomical day, should be defined for use in the calender. (e) If the lunar months have to be kept, there should be convenient devices for luni solar adjust- ments. 266 Siddhdnta Darpana All the problems have not been solved till today. The errors in calculations also had to be corrected. Hence new calenders were started in different parts of the world by the intervention of dictators like Julius Caesar, Pope Gregory Xm or a founder of religion like Mohammad, or by monarchs like Melik Shah the Seljik or Akber. Owing to historical order of development, calenders have been used for double purpose. (i) of the adjustment of the civic and ad- ministrtive life of the nation. (ii) of the regulation of the socio religious life of the people. Divisions of day : Present division of day is in 24 hours. Minute divisions of 60 each called minutes and second division again by 60 called seconds. Thus 1 mean solar day = 60 x 60 = 86,400 seconds. Division of time and angle measures by 60 was because of 30 days in a month and 12 lunar months in a year whose lowest common multiple is 60. A day has 365 but approximate multiple of 60 is 360. Hence a civil year was taken of 360 days and a circle was divided into 360°. Thus sun will move about 1° in 1 day. In India, day was divided into divisions of 60 at each step as degree is divided. Thus 1' movement is in 1 day, V movement in 1 danda, 1" movement in 1 pala and so on. Time was measured by length and direction of shadow of a pillar called gnomon. For equal time intervals, specially during night time, water clock etc were used. Improvements were done through pendulum clocks by Galileo, spring clocks 267 Corrections to Moon using balance wheel. Most occurate are quartz docks for normal use and ammonia clocks for scientific use. For practical watches of duty or shifts of work, a day was divided into 6 parts (3 parts m day time and 3 in night). After each Nerval a bell was rung In India there were 8 shifts in a day, hence the shift of 3 hours is called a 'prahara' i.e. when a bell is hit (prahara). A watchman remains on continuous duty for a prahara, hence he is called prahari. — day = 1 ghati is called so because water clock measured the time of its filling. Since it was shaped like a pitcher it is called 'ghati' (i.e. water pot). Hence watches are called 'ghadi' in India. When water clock in turned a second time it is 2 ghati = 1 muhurtta (repeated turning of water clock). Watches observed in churches were (1) Martins - last watch of night. Monk got up 2 hours before sunrise (2) Prima - at sunrise (3) Tetra - Half way between sunrise and noon - time of saying mass. (4) Sext - at noon (hence the word siesta = midday rest) (5) Nona - Mid afternoon - Hence the word noon. (6) Vespers - An hour before sunset (7) Compline - at sunset 26# Siddhanta Darpana In India mid day is 2 praharas after sunrise (i.e. 6 hours after), hence it is still called 'two pahars'. Day was divided into 12 parts in Babylone of 30 gesh (4 minutes each). In each part approximate- ly 1 sign of zodiac will rise, it is like 12 divisions of year. In India rasi was divided into two parts (like day-night divisoin of day) called 'Hora' (short of 'ahoratra' i.e. day and night) Thus there are 24 horas is a day night or 12 in day and 12 in night. This 'hora' has become hour. This was also used in Egypt and continues till today. Counting of days in a month : The ancient Iranian calender gave 30 names for each of the days of a month. It was not very popular as the list is long and difficult to remember. Hence a week of seven days was popular through out the world. Origin of week days has been explained by Varahamihira. Each hora (24 in a day is ruled by a planet. Planets are arranged in order of decreasing orbit or increasing speeds of rotation - Sard, guru, Mangala, surya (or earth), sukra, budha and Candra. In first hora of the day, lord of the day will rule. For example, Sani will rule 1st hora on sani vara. On next day ruler will be 25th planet in the order given above. Deducting 3 cycles of 7 planets, 4th planet surya will be ruler of next day i.e. 1st hora on that day. So it is called ravivara or Sunday. Next day will be 4th from surya i.e. candra or moon called somavara or monday. Rulers of days are fixed for astrological purpose, hence it has astrological origin in India Corrections to Moon 269 and west. Ancient Egyptions had a ten days week (period in which sun covers 10° or l/3rd of a rasi called Dreskana in astrology) Babylonians started a month with new moon and marked the 1st, 8th, 15th and 22nd days of the lunar month for religious festivals. This was a sort of weak of 7 days with one holiday. In Iranian calender in which 30 days had different names 8th, 15th and 23rd were called Diniparvana for religious practices. But last week in this system was of 9 or 10 days. In veda, sadaha has been mentioned, but this doesn't seem to indicate a six days week. It seems to be six extra days after 360 in a leap year called 'Gavam Ayana' every four years. The Jews reckon the days from Saturday and indicate them by numbers i.e. 1st, 2nd 7th day. Seven days week was introduced to christian world by edict of Roman emperor Constantine in 323 AD, who changed the Sabath day (Saturday for Jewish) to the Lord's day, Sunday. In India il has been first mentioned in Atharva Jyotisa and by Aryabhata. English names of week day hav€ originated from Teutonic deities which are countei parts of Roman planetary deities. Indian Childean Teutonic Roman names names dailies dailies Ravi Shamesh Sun Sun Soma Sin Moon Moon Mangala Nergal Tiu Mars Budha Nabu Woden Mercui Guru Marduk Thor Jupiter Sukra Ishtan Freya Venus Sard Ninib Saturn Saturn 270 Siddhdnta Darpana It is note worthy that functions attributed to planets by Chaldeans are same as in Indian Astrology. Ahargana or heap of days - Count of days is used all over the world from a standard epoch to calculate the mean posotion of any planet. Mean position at required time = Mean position at initial epoch + daily motion x ahargana To make a uniform standard, a French scholar, Joseph Scaliger introduced in 1582, a system known as 'Julian days' after his father Julius Scaliger. The Julian Period is 7980 years = 19 X 28 X 15 19 is length in years of the Metonic cycle 15 is length in years of the cycle of indication 28 is length in years of the solar cycle It was found by calculation that, these three cycles started together on Jan 1, 4713 B.C. Julian period and days are counted from that day and the day is completed at noon time. This is the standard for astronomical calculations now. Julian days for some important epochs is given below Date Julian day Kaliyuga 17-2-3102 BC 5,88,465 Nabonassar 26-2-747 BC 14,48,638 Fhflippi 12-11-324 B.C 16,03,398 Saka Era 15-3-78 AD 17,49,621 Diocletian 29-8-284 AD 18,25,030 Corrections to Moon 7 7 * Hejira 16-7-622 AD 19,48,440 Jezdegerd (Persian) 16-6-632 AD 19,52,063 Burmese era 21-3-638 AD 19,54,167 Newar Era 20-10-879 AD 20,42,405 Jalali Era (Iran) 15-3-1079 AD 21,15,236 In India, siddhanta jyotisa uses ahargana from creation after wfeich 6 manus of 71 yuga each have passed, in current 7th manu 27 yuga have passed. In 28th yuga, Satyuga, Treta and divapara have passed. Present kali yuga started on 17-2-3102 B.C. Ujjain midnight. In this kali yuga is 4,32,000 years. Dvapara, Treta, Satya yuga are 2, 3, 4 times. A yuga is 10 times kali = 43,20,000 years. Before each manu there is a sandhya of a satyayuga. Thus years from creation till beginning of kali yuga are 1, 97, 29, 44, 000 years. To find the ahargana for calculation, we deduct the years spent in creation = 47,400 divya years x 360 solar years. After this period all planets started from zero position which is called epoch. Ahargana at beginning of kaliyuga is 714, 402, 296, 628 Tantra granthas count the ahargana from kali era. Each karana book has used its own epoch. In present calculations Jan 1,1900 is important epoch. For this day Julian days are 2,415,021 and kali ahargana are 1, 826,556. (7) Solar calenders in History - (a) Egyptian calender - This has 12 months of 30 days each, starting from Thoth on 29th August as per Julian calender. This was old religious calender, hence extra 5 days were attached 272 Siddhanta Darpana in the end which not part of any month. Since the year was short by 1/4 days from 365-1/4 days,] the heliacal rising of Sirius star would re-appear i at the beginning of year after 1460 years. This was^ called Sothic cycle as Sothis (Isis) was the goddess of sirius. In 22 B.C. the year started on 29th August the Pharoahs (kings) of Egypt tried to introduce leap year, but this never became popular. Ptolemy in 238 introduced a leap year, but old calender also continued side by side. Egyptians did not use any continuous era, but counted the number of years of each reign separately. For astronomical purposes, Nabonassar Era was used in Babylone. This was used as a reference by all countries for sumplicity (b) The Iranian calendar - Around 520 B.C. Darius introduced a solar calender like Egyptian with 365 days each. It had 12 months of 30 days each and each day had a specific name. The names are similar to vedic names. 5 days extra were attached in the end. Adjustment of 1/4 extra day each year was done by adding a month of 30 days in a cycle of 120 years. From 16-3-1079 A.D, Seljuc sultan Jelaluddin Malik Shah introduced a new calender Tarikh-e* Jalali, starting from 10th Ramadan of Hejira 471. It was 365 days year with 8 .intercalary days in 33 years. The year started from vernal equinox day or next day. Its lenght was 365.242 42 days. Riza Shah Pahlavi introduced a strictly solar year and restored the old Persian names of month; in use before Darius. The year started from 21 or 22 march. First 6 months were of 31 days each. Corrections to Moon 273 Last month was of 29 days or 30 days in a leap year. Roman calendar (Christian Calendar) - The so called Christian calendar had nothing to do with Christianity. It was originally the calender of semi savage tribes of Northern Europe, who started their year some time before the beginning of spring (March 1 to 25) and had only 10 months of 304 days, ending about the time of winter solstice (December 25). The remaining 61 days formed a period of hybernation when no work could be done due to on set of winter, and were not counted at all. This calender was adopted by city state of Rome and some modifications were made. Second Roman king of legendary period Numa Pompilius added two months (51 days) to the year in about 673 B.C. making a total of 355 days. January (named after god Janus who faced both ways) and February were added in beginning and March became the 3rd month now. Number of days became now 29,28,31,29,31,29,31,29,29,31,29,29. Adjustment of the year to the proper season was done by intercalation of a thirteen month of 22 or 23 days (caDed Mercedonius) after two or three years between February and March, the extra month was actually 27 or 28 days but, the last 5 days of February due to be repeated after extra month, were not repeated. The correction at alternate year could have given 45 (22+23) days in 4 years pr 11-1/4 days on average. Thus it made a year only one day longer than 365-1/4 days. But this was irregular and caused a lot of discrepancy from the seasons. 274 Siddhanta Darpana Julius caesar, on his conquest of Egypt in 44 B.C. was advised by Egyptism astronmer Sosigenes that mean length of year should be jlS-i/l days.Normal length should be 365 days and one extra day should be added every fourth year. The*! the fifth month from March, Quintilis was changed to July (Julius) in 44 B.C. in honour of Julius Caeser and length of months were fixed at their present duration. Extra leap year was obtained by repeating the sixth day before kalends (first day) of March. In 8 B.C., sixth month after March, Sextilis .was changed to August in honour of Augustus, successor to Caesar. To correct the seasons, §Q days were added to 46 B.C. 23 days after February and 67 days between November and December. Thfe year of 445 days was known as year of confusion. Caesar wanted to start the new year on 25th December, the winter solstice day. But people resisted, because new moon was due on January 1,45 B.C. Caesar had to accept the traditional landmark of the year. Weekdays of 7 days week were introduced sometimes in 1st century AD on pattern of chaldean astronomers. Days of crucification of christ and his ascending to heaven was fixed arbitrarily on Friday and Sunday later on. New Testament only says that he was crucified on a day before Passoyttr festival of Hebrews which was on full moon day of the month of Nissan. The present christian era started at about 530 AD. When era beginning was fixed from the birfli year of christ, birth day of christ was fixed oil December 25, which was winter solstice day an!8 ceremonial birth day of Persian god Mithra in 1st Corrections to Moon 275 century B.C. However, a Roman inscription at Ankara shows that king Herod of Bible who had ordered massacre of children after birth of christ, was dead for 4 years at 1 AD. Therefore, christ must have been born before 4 B.C. The Julian year of 365.25 days was longer than the true year of 365.2422 days by 0.00788 days, so the winter solstice day which fell on 21 December in 323 AD, fell back by 10 days in 1582 AD. In 1572, Pope called a meeting to discuss the correction. In 1582 Pope Gregory XIII, published a bill instituting a revised calender. Friday, October 5 of that year was to be counted as Friday, October 15. The century years which were not divisible by 4 were not to count as leap years. Thus the number of leap years in 400 years was reduced from 100 to 97. length of years was 365.2425 days, the error being only one day in 3300 years. This was adopted immediately by the catholic states of Europe. But Britain adopted it in 1752, China in 1912, Russia in 1918, Greece in 1924 and Turkey in 1927. Revised rules for easter have not been adopted by the Greek Orthodox church. World calender : To remove the working defects of Gregorian calender, a world calender was propsoed to UNO in Geneva meetign of ECOSOC in 1954. In this calender week days of every year are same. One extra week day in 365 days is kept after 30th December called W or world holiday. In leap year another world day was to be introduced after 30th June. Every year was same for counting of week days. Each quarters of 3 Months was of 91 days, 13 weeks. First month of each quarter was 31 days and remaining of 30 276 Siddhanta Darpm days. So each quarter has same form of calender! Each year (each quarter also) begins on sunday| Each month has 26 working days, plus Sundays. J (8) Luni Solar Calenders - We need very accurate measurements and] complicated procedure to tally lunar and solarj calenders. Mean lunar synodic month = 29.53058*1 days = 29 d 12h 44 m 2s/ with a variation of ± 7 hours Mean sidereal period of moon = 27.3216611 days = 27d 7 h 43m 11.5s. 1 with a variation of ± 3— hours. 2 12 lunations (synodic) amount to 354.36706 days while tropical solar year is 365.24220 days. Length of lunar year is shorter by 10.87514 days, and there are 12.36827 lunar months in a solar year. Tropical solar year is varying very slowly and is becomig shorter by 8.6 seconds == .0001 days in 1600 years. Thus at kali beginning or in Sumerian times it was 365.2422 days. All ancient nations had almost accurate knowledge of the mean synodic month. However, no rules could be fixed for tallying the lunar year with solar year. Hammurabi (1800 B.C.), law giver king of Babylonia, has a record saying that the thirteenth (extra) month was proclaimed by royal order throughout the empire on advice of priests. Practically the start of first month was adjusted with ripening of wheat. Corrections to Moon 277 * Later Babylonians, called Chaldeans around 600 B.C. fixed some empirical relations in lunar and solar years for correction of calender in form m lunar months = n solar years. where m and n are integers Some convenient periods were Octaeteris - 8 tropical years = 2921.94 days 99 lunar months = 2923.53 days. This gave 3 intercalary months in 8 years with error of only 1.59 days. In about 500 BC (383 B.C. according to father Kugler) 19 year or Saros cycle was used with 7 intercalary months 19 solar years = 6939.60 days 235 lunar months 6939.69 days This gives a discrepancy of 0.09 days in 19 years or of 1 day in 209 years. Their 19 years cycle was of 6940 days with leap years on 1st, 4th, 7th, 9th, 12th and 15th year in the first month and in 18th year at 7th month. First month started with 30 days, then other months were atternately 29 and 30 days. Thus a normal year was of 354 days, but in 5 years of 19 year era one extra day was added to last month, making the year of 355 days. After adding intercalary year, the year was of 354, 355, 383 or 384 days duration. Effect of this arrangement was that the first month Nisannu start was never more than 30 days away from vernal equinox. The Chaldeans used gnomon for ascertaining time of 2 e quinoxes and 2 solstices which divide a solar year mto 4 almost equal seasons. 278 Siddhanta Darpam Eras of Western world - Dated records of kings in Babylon beings from about 1700 B.C. (Kassite kings). In Egypt also regnal years were used. But in Babylon, months and dates were of lunar month is while they were solar in Egypt. Hipparchus (140 BC) and Ptolemy (150 AD) of Greece used the records of systematic observa- tions of Babylone from 747 B.C. since the time of one king ftabu Nazir. Though they counted the astronomical era from 26 Feb. 747 B.C. in that reign, they adopted Egyptian solar years of 365 days each for ease in calculation of dates. Macedonian Greek had their own months, but after they settled in Babylon in 313 B.C, they adopted their months to Chaldean months, 1st month Dios starting with 7th month of Chaldeans at autumnal equinox. Seleucus, a general of Alexander, a, Macedonian Greek founded a big empire in west Asia and started his own era Seleucidean era. Iri official or Macedonian reckoning it started from the lunar month of Dios near autumnal equinox in (-311 AD) or 312 B.C, with greak month names. In Babylonian reckoning, the months had Chaldeari names starting from Nisan near vernal equniox. Parthian era was started in 248 B.C. when Persia] again became independent empire. | Ancient Jewish calender was lunar and theii| month names are derived from Chaldean narnei or vice versa. The day began in evening an<| probably at sunset. Extra month was added whe^ necessary by making two months of the last montt| Adar - original was named veadar followed bj$ Corrections to Moon 279 Adar. Year beginning was changed from Nisan month to Tisri corresponding to Mecedonian month of Dios. Around 4th century A.D. rules were formed for intercalation. In a cycle of 19 years 3,6,8,11,14,17 and 19th years had extra month. Start of first months was adjusted, so that week days of important festivals do not change. Thus a common year could have 353, 354 or 355 days and a leap year of 383,384 or 385 days. 10 of the middle months had got fixed duration of 29 or 30 days. Extra month was of 30 days. The other two (1st and 12th months) varied according to length of the year. Jewish era is called Anno Mundi or libriath olum or Era of Creation or Freedom. According to mnemonic Beharad, this era* is supposed to begin at the beginning of lunar cycle on the night between Sunday and Monday, Oct 7, 3761 B.C., at 11 hours 11-1/3 minutes PM. (Be = Beth i.e. 2nd day of week), ha (he = five, i.e. fifth hour after sunset) and Rad (Resh) delet i.e. 204 minims after the hour, 18 minim = 1 minute) In Bible, eras have been mentioned from flood, exodus, the earthquake in the days of king Uzziah, the regnal years of monarchs and Babylonian exile. After exile, they counted years from Persian kings, and then from Seleucid era. Days have also been counted from fall of the second temple. 312 - Seleucidean era = Christian era B.C. (Jan to Sept) Saleucidean era - 311 = Christian era AD (Jan to Sept) Year 1 after destruction of second temple 280 Siddhanta Darpana = 3831 Anno Mundi = 383 Seleucid = 71 A.D. Islamic Calendar - This is purely lunar calender now and has no connection with solar year. The year consists of 12 lunar months; beginning of each month is deter- mined by 1st observation of crescent moon in the evening sky. The months have 29 or 30 days and the year 354 or 355 days. The new year day of Islamic calendar loses about 1 month in 3 years, and completes the retrograde cycle of seasons in 1 , 32 — solar years. Hejira (A.H.) was introduced by caliph Umar about 638-639 AD, stating from evening of 622 AD, July 15, Thursday (Since sunset Friday started in Islamic calender). Then crescent moon of the 1st month Muharram was first visible. This was the new year day preceding the emigration of Muham- mad from Mecca (about Sept 20, 622 AD.). The months are alternately of 30 and 29 days from 1st month. Last month is 29 days in normal year and 30 days in a leap year. If Hejira year is divided by 30 and remainder is 2,5,7,10,13,16,18,21,24,26 or 29 then it is a leap year. Thus 11 leap years in 30 years, gives the cycle of 10,631 days which is 0.012 days less than the true value. Dr. Hashim Amir Ali of Osmania University has showed that the mohamadan calender was originally luni-solar. Upto the last year of the life of Mohamad; i.e. upto AH 10 or 632 AD, a thirteenth month was intercalated when necessary. The family of astronomers, known as Qalamas Corrections to Moon 281 decided at hajj in last month, whether 13th month will be added or not. This should have been 3 times in 8 years or 7 times in 19 years, but use of discretion by eldest Qalama created confusion afterwards. Thus AH 11, a normal year started on 29th March 632 AD. after vernal equinox. Thus all the previous years with intercalation, started after sighting new moon after vernal equinox. Thus the initial epoch of Hejira era was at the evening of March 19,622 AD, Friday, the day following the vernal equinox. Names of Lunar Months Indian Chaldean Mecedonian Jewish Islamic Caitra Addaru Xanthicos — — Vaisakha Nisannu (30) Artemesios Nissan Muharram Jyestha Airu (29) Daisies lyyar Safar Asadha 4 t Sivannu (30) Panemos Sivan Rabi-ul-awwal Sravana Duzu (29) Loios Tammuz Rabi-uls-sani Bhadra Abu (30) Gorpiaios Ab Jamada alawwal Asvina Ululu (29) Hyperberetrios Ellul Jamada as sard Karttika Tasritu (30) Dios Tisn Rajab Margasirsa Arah/Samnah(29) Appelaios Marheshvan Shaban Kisilibu (30) Audinaios Kisilev Ramadan Pausa • Dhabitu (29) Peritios Tebeth Shawal Magha Shabat (30) Dystros Shebat zil kada Phalguna Addaru (29) Xanthicos Adar and Zil hijja Caitra Veadar (9) Old Indian Calendars : A. Vedic Calender - Vedic calender was luni solar. Year was named in three manners - Solar year, civil year and lunar year (normal and intercalary). Sama = Fixed year or constant. It is opposite to 'masa' i.e. formal of 12 masa of 30 days each. Thus it means a year of 360 civil days or 365 solar days (i.e. 365-1/4 days) 252 Siddhdnta Darpana Lunar years are called vatsara - which are of 5 types- Samvatsara, anuvatsara, Parivatsara, Id- vatsara and Idavatsara. Anuvatsara is also called Iduvatsara. When these indicate a sequence of 5 solar years of 366 days each, vatsara is a sixth year of 360 civil days or sama (as per yajus jyotisa). Names of thirteen months in Taittiriya Brahmana (3-10-1) are Aruiia, Aruna rajas, Pundarika, Visvajlta, Abhijit, Ardra, Pinvamana, Annavan, Rasavan, Ira van, Sarvan sadha, Sambhar and Mahasvan, Mahasvan appears to be increased month (with extra days in a solar year). 6 seasons of two solar months each are as follows - 1. Vasanta - Madhu and Madhava 2. Grisma - Sukra and Suci * 3. Varsa - Nabhas and Nabhasya 4. Sarad - Isa and urja 5. Hemanta - Sahas and Sahasya 6. Sisira - Tapas and Tapasya Taittiriya Brahmana has given a list of 24 half months (1 fortnight), names of day times and night times in sukla and krsna paksas - 60 names, names of 15 muhurtas in sukla paksa day and night, Krsna paksa day and night - 60 names and 15 parts of each muhurta / called prati muhurta). Name of lunar months were named after the naksatras entered by moon on purnima day. Rk veda (1-15-1) tells that^Indra drinks soma juice with seasonal adityas on full moon day. Thus Indra is always at a point 180° away from sun. Corrections td Moon 2S ^ Aditya corresponding to different seasons are (1) Mitra - sisira (2) Aryaman - vasanta (3) Bhaga - grisma (4) Varuna - Varsa (5) Daksa or Dhata - sarad (6) Arhsa - Hemanta Rk veda verse 10-72-4 by Sunahsepa gives method of deciding about inclusion of intercalary month - Daksa was born of Aditi and Aditi was Daksa's child. The whole ecliptic was Aditi and its division were adityas - 6 for each season, 12 for each month or 13th for extra month. First point of Daksa division was the start of ecliptic zero degree. Year started with rise of this point on eastern horizon with sun. When the next rise was not before 13th full moon, 13th month was extra month otherwise it was month of next year. In santipatha also it is stated — In Vajasaneyi samhita, two adhika masa are named. Sansarpa is extra month before winter solstice. Another is malimluca. Ksayamasa (lost month) was' called Arhhaspati. (Yajur-VS, 22-30) In a solar year of 365-1/4 days, 5 or occasional- ly six days are extra after civil year of 360 days. These have been called atiratra (i.e. extra days after grand night). Taittiriya Samhita (7-1-10) says that 4 atiratra make the year incomplete, while 6 atiratra give excess, so five are the best. Aitareya Brahmana has denned Tithi as the time during which Moon sets and rises again (32-10). Thus like civil day from sunrise to sunrise, tithi is a moon day from moon rise to moon rise. 284 Siddkanta Darpana r. In sukla paksa tithi was from moon set to moon set, and in other it was moon rise to moon rise. Atharva Vedanga jyotisa has defined two karana in each tithi - one from moon rise to moon set and second from moon set to moon rise. These tithi and karana were of unequal length. Later on they were made of equal length defined on basis of moon phase. B. Vedanga Jyotisa (Rk veda) : This is described in only 36 verses in anustupa chanda including introduction and importance*. This is one of the six parts of Rk veda. Though it is shortest, it gives a most comprehensive, luni solar calendar so far. It was written by 'Lagagha' whose place was 35° N latitude, nothern border of Kashmira, may be the present town of Alma-Ata of Kyrghiz! This place might have been first place of learning^ hence first school is called alma-meter. Efforts to explain its meaning on basis of 5 years cycle (yuga) were unsuccessful, by various anthors as B.G. Tilak, S. B. Diksita and T.S.K. Shastri, R. Shamshastri etc. Tanca samvatsara mayam yugam' was interprated that a yuga has 5 years (meaning of samvatsara). But samvatsara is one of the 5 types of lunar years and its meaning should be - A yuga has 5 years of samvatsara type, remaining years of other 4 types. If calculations are made on that basis, a yuga has 19 years, with 5 types of vatsaras, out of which 5 are sanvatsaras. This also gives meaning of other types of years. This gives correspondance of solar and lunar years in terms of tithis, days and naksatras also. Corrections to Moon 285 Time cycle : There are 360 tithis in a lunar year. Solar year is bigger by 10.89 days. With reasonable accuracy, 7 intercalary months (adhikamasa) occur in a cycle of 19 years. Thus 228 solar months = 235 lunar months Addi- tional 7 months form 7 x 30 = 210 tithis. Thus there is difference of — — = 11— tithis (10.89 days) 19 19 v J ' between a solar and a lunar year. Thus a solar year consists of 371 — tithis. If we assume a leap year in a cycle, we have 18 years with 371 tithis and one year (i.e leap year) with 372 tithis. This cycle of 19 years is called a yuga. Calculation of Rtu sesa - A year has 12 months (lunar) each having two parts sukla paksa (called, sudi - i.e. Suksma diwas - or suddha diwas) and Krsna paksa (Badi i.e. bahula diwas, extra days). Thus 24 paksa of a year have difference of 11 tithis from solar year. Difference in each paksa is — = 0.458 = — tithi approximately Thus a lunar paksa = 15 tithis Solar paksa = 15 — tithis For calculating extra tithis for each half of solar year, we have to add 11/2 tithis = 11 karanas. Thus we have to subs tract 1 karana after half year from 12 karanas got after taking 1 karana for each paksa approximately. One tithi is taken extra in a completed yuga of 19 years. When cumulative total of extra karana after a semester is more than 60 286 * Siddhanta Darpana karanas = 30 tithi = 1 month, one extra month is added in that semester. Classification of years : If one karana is not dropped in each semester, then Rtusesa will be 12 tithis per year or 6 tithis per ayana (semester). Thus start of ayanas will be after 6 tithis each and after 5 ayanas (2-1/2 years) the cycle will be complete and one extra month will be added so that the month starts again with Magna sukla 1 (1st year of month). Thus years were classified according to range of tithis on which 1st day of year fell - Samvatsara - Sukla 1 to 6th Anuvatsara - Sukla 7th to 12th Parivatsara - Sakla 13th to 18th (or badi 3rd) Idvatsara - Badi 4th* to 9th Ida vatsara - Badi 10th to 15th When we decide adhikamasa for each lump of 60 karana Rtusesa, 5 years in 19 years yuga are of samvatsara type. 3 years are Idavatsara, lagging behind most, hence the adhikamasa occurs in 1st semester of those years (6th, 9th and 17th). which can be seen by calculation. Four years are of Idvatsara type lagging 18-24 tithis, hence the adhika masa is added in 2nd semester of (3rd, 11th, 14th and 19th years) Naksatra calculation for sidereal lunar year - In a lunar month, moon completes its circular journey of 27 naksatras and travels about 2 naksatras more. To be more accurate, it completes 13 revolutions in 12 lunar months. Thus in 19 solar Corrections to Moon 287 years, there are 254 sidereal months and in 19 lunar year 19X13 = 247 sidereal months of moon. Thus difference in 2 cycles is 7 sidereal months = 7 x 27 = 189 naksatras. Thus we have 10 naksatras per year for 18 years and 9 extra naksatra in one year 1 solar year = 361 naksatra Leap year = 360 naksatras 1 lunar year == 351 naksatras ( = 13 x 27) 10 Solar semester - lunar semester = — = 5 naksatras. Rtuses in terms of naksatras is calculated by assuming a total of 190 naksatra for 38 semester i.e. 5 for each. To be more accurate it is 5- — naksatra. Thus calculation is to be started from sravistha (Sravana + 38 parts of l/38th i.e. complete dhanistha). Complete naksatra divisions will be counted from Sravana at interval of 5 naksatras each for every ayana (semester). Thus list of naksatras is given at intervals of 5 naksatras indicating only one letter from each naksatra. This verse was deciphered brilliantly by Sri S.B. Diksita. According to moon naksatra at start of year also years can be classified. Thus samvatsara can start from sravana upto 5.4 naksatra (Asvinl) - 2.7 difference for each semester. Anuvatsara is Asvini (5.4) to Ardra (10.8) - countings done from sravana. Parivatsara can start up to 16.2 (uttaraphalguni) Idvatsara upto (21.6) anuradha and Idavatsara in remaining naksatras. C. Visvamitra's astronomy - also indicates a 19 year yuga. His hymn in Rk veda III - 9-9 reajds '■■-31 4 288 Siddhanta Darpana i.e. 3339 devas (dyues or parts of ecliptic = aditi) worshipped agni (sun or krttika naksatra) by rotations in the sky. This is based on calculation of solar naksatras for each parva (or paksa). First solar year of 372 tithis is the basic year in which sun crosses the 27 naksatras. Thus 1 solar naksatra = 372/27 = 124/9 tithis. To avoid fractions, angular distance travelled i by sun in a tithi is divided into 9 parts called 5 'Bha-amsas' or bhansa. (Bh°) 1 Tithi = 9 Bh° of sun f 1 Naksatra = 124 Bh ° j 13° 20' 1 1 Bh° = = 6'27".l approx | 1 parva = 15 tithi = 135 Bh° = 1 Naksatra + j 11 Bh° In 1 Ayana = 12 parva i = 12 (Naksatra + 11 Bh°) = 12 Naksatra + 132 1 Bh ° = 13 Naksatra + 8 Bh° Thus 8 Bh° arise in one parva (in addition to j completed naksatras). In 371 tithis of a solar year there are 371x9 * Bh° = 3339 Bhansas, which are indicated by Visvamitra. Saros cycle of Chaldea was of 18 years and 10.5 days after which eclipses are repeated. 3339 tithis = 111 synodic months + 9 tithis. Corrections to Moon 289 This is half of Saros cycle of 223 synodic months 3339 /^.n \ synodic years = 3240 1080 = 1080 3 sideral years are a yuga x (1/3 of Visvamitra's mahayuga). This is the period of precession of equinoxes for 1 parva (15 tithi or 15° movement of sun in 15 x 72 = 1080 years). A day is divided into 603 kalas = 30 muhurta. Moon crosses 1 naksatra in 610 Kalas. D. Yajus Jyotisa : This is part of Yajurveda whose commentary by Somakara was available. It has 44 verses out of which 30 are common with Rk jyotisa Due to that reason, scholars have tried to combine these two into one text of 50 verses and interprate both on basis of 5 years yuga. However, Rk has 19 years yuga and yajus has 5 years yuga. This has been specified in verse 31 of this text .-'.-. (In a yuga) there are 61 savana months, 62 lunations and 67 sidereal months i.e. naksatra masas; 30 days make one savana month and 30-1/2 days make one solar month. This along with verse 4 tells that we have one adhika masa after every 2-1/2 years - clearly specify a 5 year yuga. This is the different meaning of this text. Adjustment of luni-solar years in this system can be done on basis of two statements in contemporary texts. Mahabharata, santi parva ch 301 tells. SR «'<4<tK]U|i I ^*TOHf ^ m tot (By way of leap) drop years as well as months. 290 Siddhanta Darpan Order of leap years is indicated by Taittirlya Brahmana (part of yajurveda) (3-10-4) which gives list of years. (1) Sarhvatsara (2) Parivatsara (3) Iduvatsara which is same as anuvatsara according to Madhava (4) Idvatsara (5) Idavatsara and (6) Vatsara : This extra sixth vatsara is a sama or savana year of 12 savana months (30 days each according to verse 31) and comes after each yuga of 5 years in a yuga in the above order. Thus each of the five years is of 366 days and sixth year of 360 days balance the extra days counted in the yuga. The year after sama or vatsara may not be samvatsara, it will be decided according to tithi or naksatra at beginning of year as defined in Rk jyotisa. Thus the ommitted years can be thought as dropped years or ksaya years. Thus in a cycle of 5 yuga% of 25 years we drop six years including 3 leap years whose adhikamasa also gets dropped in the process. Thus we get 19 year yuga as before. But we get a simpler 5 year yuga. There is a difference of 4 hours 23 minutes between vedanga cycle of 19 years and 19 solar sideral years. It we acid 8 years at the end of eight 19 years yugas, we get 160 years. The difference at that end reduces to 23 minutes only. Time period - RVJ verse 5 tells that year started on full moon day in Magna in winter season when sun was in vasava nakshatra (1st year of yuga) YVJ verse 6 tells that sun in beginning of sravistha indicated beginning of Magna and sun <3 Corrections to Moon 291 * in mid point of aslesa started beginning of south solstice. Assuming 1 ° precession in 72 years, this indicates Rk jyotisa in 2976 BC and yajus jyotisa in 2352 BC. However verse 3 of RVJ indicates that the theory was coming since long when these verses were composed. E. Gavlm Ayana has been mentioned in Tattiriya samhita, Satapath brahmana, Gopatha brahmana and Baudhayana srauta surra. It indicates a 4 year yuga with 1 leap year according to Prof. R. Shama Sastri (1908 - gavam ayana) Accumulation of l/4th day of each of previous 3 years combined with 4th year to make one extra day like the Julian calender. Thus this is a cow with 4 legs or 3 parents of sun. Four years of this yuga were called kali, dwapara, treta and krta yuga. These are also called 1st, 2nd, 3rd and complete (krta). Krta is also called Satya or Rta i.e. which really came (as a full day) If year or yuga starts in evening, 1st year (kali) will end at midnight after 365-1/4 days (sleeping time). 2nd year dvapara will end in morning (rising time) on 366th day. 3rd year Treta will end on 366th day noon, when sun is at highest. 4th year krta or satya will end in evening when people are moving. Thus Aitareya Brahmana tells - sleeping is kali, rising is dvapara, standing is treta and moving is krta, so keep on moving (7-15). This is attributed to Manu. This was around 23,720 B.C. as Taittirlya samhita indicates (7-4-8) vasanta at phalguna full moon. Rk veda indicates rains in Mrgasira naksatra indicating same time. Thus 4 years yuga with 4th as leap year appears to be first system started around 24000 B -C. Then 19 years yuga with 7 leap years (lunar) 292 Siddhanta Darpaqm 4 'M W\ 4 of vedanga jyotisa continued upto about 3000 B.C i.e. kali beginning. With kali era smaller 5 year yuga of yajurveda forming a 19 year yuga was started. F. Jaina calendar - Surya prajnapti and Candra prajnapti are two principal texts written at the time of Mahavira about 600 B.C. However Jain Tir- thankars and their astronomical traditions might have started along with yajurveda or early brahmana texts. 1 * '<: It There are five kinds of sarhvatsaras (years) «• 1 (1) naksatra samvatsara (2) yuga (cycle) sam-| vatsara (3) Pramana (standard) samvatsara (4)| Laksana (symptomatic) samvatsara and (5) sanlcara (saturn) samvatsara Days in year Nakstrika Lunar Rtu Solar Abhivar- dhana 327-51/67 days 354-12/62 360 360 382-44/62 Months in year 27-21/27 29-32/62 30 30-31/62 31-121/124 1 Month of 5 year cyde 67 62 61 60 57-3/13 l ■■„( A five year yuga consisted of 5 lunar samvatsaras with 3rd and 5th years having extra; months called abhivardhan samvatsara. This i$ almost like yajus jyotisa but simpler. Naksatra samvatsara was named according to naksatra occupied by Jupiter at the time of complation of samvatsara - these are same as present months names of India. Sanlcara samvatsara was time of sard in crossing 1 naksatra out of 28 with mean motion. Corrections to Moon 293 (10) Indian Eras : (A) The eras started after kali era are based on concept of Mahayuga of 43,20,000 years or a kalpa of 1000 mahayuga. Yuga concept is attributed to Aryabhata in 499 AD and kalpa concept to Brahmagupta in 627 AD. However, Brahmagupta refers to Visnudharmottara Purana. Smrtis also have been referred to. Aryabhata himself has followed purana tradition, except to treat four parts of yuga as equal. The five limbs of calendar, known as pancanga have already been explained in the previous chapter. Their brief definitions are given again - ° (1) Vara - Runing weakday in a cycle of seven days. (2) Naksatra - Naksatra occupied by moon. This almost means one day and was most popular hi. mahabharata era to indicate a day (in Valmiki Ramayana also). (3) Tithi - Moon rise to moon rise system was changed. Vedanga jyotisa started equal division of titrus depending on phases of moon. Tith* = M° on - sun 12° pvf. S U ° ti f ntS above 15 indicat e krsna paksa and «ctra days beyond 15th are counted as tithi number. £ence Krsna tithi is called' bahula divas' (extra ^u° r ' badr " short Sukla tithi ^ called sudi - suddha divasa. rko / 4) ****** is half of tithi. In veda it was moon se to moon set or vice versa. From vedanga jyotisa lt is exactly half of tithi. * W • 294 Siddhdnta Darpctr0 Moon - sun Karana = — 6 A Corresponding to 1 day = 2 karanas extra at I beginning of 19 yeear vaidika yuga, one karana at | each end of amavasya are omitted from running j cycle of seven karanas. These 4 are fixed karanas. j Remaining 56 karanas start from sukla 1st tithi 2nd| half in which seven karanas are repeated 8 times I in a month. j (5) Yoga - It is only a mathematical concept. 1 It means sum of longitudes of sun and moon and A one cycle of 360° makes 27 yogas. Originally there! were only 2 yogas. Vyatipata was when kranti of | sun and moon were equal but their longitude was j equal and in opposite directions. When longitude '■ is same but kranti is equal and opposite, it was ! called 'vaidhrti'. Thus yoga was means to calculate" these and subsequently others were included to make a complete cycle of 27 yogas like 27 naksatras. There is another kind of yoga which is combination of vara, tithi or naksatra for auspicious works. B. Rules for calender - (1) A lunar month stars from Sukla 1 (called ananta) or from Krsna 1 tithi called Purnimanta. Lunar month is named after the naksatra ap- proximately occupied by moon on purnima of that month. Amanta is called mukhya and other gauna. (2) A solar month starts with entry of madhya surya in a nirayana rasi (i.e. fixed point of zodiac). The first day of month may start on same day, Corrections to Moon 295 next day or 3rd day according to occurrance of sankranti in different parts of day or night. (3) In luni-solar year a lunar year tallies with a particular sankranti of surya every year. In a lunar month when there is no sankranti the month is called adhika masa. The month having two sanknantis, is called ksaya masa, the month corresponding to 2nd sankranti of the month is dropped. Ksaya masa is called amhaspati. The adhika masa before it is called sansarpa (or in 1st ayana uttarayana of the year). The adhika masa after ksaya masa or in 2nd ayana is called malimluca. (4) Uttarayana starts when sun starts its northward journey after winter solstice or sayana makara samkranti (24th or 25th December). Daksinayana starts when sun starts going south from summer solstice i.e. sayana mithuna sankranti (26 June). A year may start with start of uttarayana as in vedanga jyotisa or from equinox in uttarayana - vernal equinox which is middle point of uttarayana. Instead of exact equinox point of uttarayana, we count entry into fixed zodiac rasi which follows 23 days later at present. C. Rules for sankranti (1) In Orissa, solar month begins on same day as sankranti of madhyama surya, irrespective of the part of day (sun rise to sun rise) it falls in. (2) Tamil rule - If sankranti takes place before sunset the solar month begins on same day, otherwise from next day. (3) Malabara rule - In parahita system of Kerala, if sankranti takes place before lapse of 3/5th 296 Siddhdnta Darpana of duration of day (i.e. about 18 ghati or 7h / 12m after sunrise - about 1-12 p.m.), month starts on the same day, otherwise from next day. (4) Bengal rule - When sankranti takes place before midnight, month starts from next day, if it is after midnight, then from third day (next to next day). If sankranti is within 1 ghati of mid night, i.e. 24 minutes before or after, tithi at sunrise time is examined. If sankranti is before lapse of that tithi then month starts on next day. If it is after tithi then from 3rd day. For karka and makara sankranti this rule is not followed. D. Eras started in India - Eras of long period have been described in all old civilisations. They describe three great floods. After one great flood Brahma appeared and started the civilization. Then saptarsis were born, rule of Daitya, Deva and Danava followed. That may be called Deva yuga. Devayuga ended with another great flood in which rudiments of life were preserved by Manu (or Nuh). After re-settlement human eras began. These were formed into a cycle of 12,000 divya years, l/10th was kaliyuga, 2, 3, 4 times were dvapara, treta and krtayuga (or satyayuga). There was another era called sap tar si era which is equal to 2700 divya versa, assuming that they remain in one nakSatra for 100 such, years. Another count in Vayupurana mentions saptarsi yuga as 3030 manusa varsa. Thus divya varsa appears to be solar sidereal year of 365-1/4 days and manusa varsa is sidereal lunar year of 327.4 Corrections to Moon 297 days. These values give the above ratio of values given in Vayu pur an a. Each yuga was further divided into sub parts, like Vayu purana indicates 24 parts of treta yuga and great personalities have been named in each part. Each part considered equal, parts of Treta were of 150 years each. Dvapara had 28 parts = 2400 solar years = 85.7 years approx* It is more convenient to keep 4 parts as sandhi periods after treta and after dvapara. Then each part is of 1 century, i.e. 1 naksatra of saptarsi. Some cor- roborating quotations for this time scale are - (1) Megasthenese quoted by Pliny (Indika of Arian ch IX) - From the days of Father Bacchus to Alexander the Great, their (Indian) kings are reckoned at 154 whose reigns extend over 6451 years and 3 months (Pliny } • Father Bacchus was the first who invaded India and was the first of all who triumphed over the vanquished Indians. From him to Alexander the Great, 6451 years 3 months. ... .reign by 153 kings in intermediate period (Solin) From the time tb Dionyson (or Bacchus) to Sandrokottos> the Indians counted 153 kings and a period of 6042 years. Among these a republic was thrice established, another for 300 and 120 years. Note - Bacchus is mentioned in Bible, becomes Dionyson in Greek. This is derived from 'Danusunu' (son of Danu third wife of Kasyapa — or danava) or Vipratitti (Bacchus). Herodotus has stated that Bacchus was called Orotol in old Arabic. This is derived from Vipratitti. * Siddhanta Darpan* (2) Among sons of Kasyapa prajapati, eldest were born from Diti called Daitya. Next were from Aditi called Aditya or deva. Daitya were earlier and first to rule over world, hence they were called 'purvadeva'. Last were born from youngest wife Danu called danava. Daitya and danava were called asura and they were anti to deva or Sura. Herodotus writes on basis of Egyptian priests (part 1, p. 136) The twelve gods were, they affirm, produced from the eight and of these twelve Hercules is one. Hercules belongs to the second class, which consists of twelve gods and Bacchus belongs to the gods of the third order (P. 199). Note - Hercules is derived from 'sura kulesa' i.e. Visnu. One form of Visnu was vamana, youngest of the twelve adityas, who conquered Bali. n Daitya Hiranya kasipu = Zeus Prahlada = Epaphos = Libye Virocana = Beor Bali = Bala = Bel = Baalim I — y Bana Candrama = Cadmus Greek names are according to Pedigree by Nounos (1-377), Bible Duternomy 23-4 tells - they hired against thee Balaam the son of Beor of Pethor of Mesopotamia. In Jesus 3/7 and 6/28,30, it is called Baalum and Baal. Corrections to Moon 299 Devayuga before krta yuga has been men- tioned in Ramayana Bala kanda 9/2 and Jaimini Brahmana 2/75, Mahabharata Adiparva 14/5 Sabhaparva 11/1, Vanaparva 92/7. (3) Floods - Encyclopaedia of Religion and Ethics - Article on ages - The cuneiform texts mention kings before the flood in opposition to kings after the flood. In times before the flood, there lived the heroes, who (Gilgames Epic) well in the under world, or like the Babylonian Noah, are removed into the heavenly world. At that time, there lived, too, the (seven) sages. Berosus, priest of Marduk temple of Babylon under rule of Selucus writes - There were 86 kings after flood in first family who ruled for 34,090 years. Then 5 more families ruled one after the other. It is note worthy that among the south Amercian Indians, it is generally held that the world has already been destroyed twice, once by fire and again by flood; as among the eastern Tupies and Aravaks of Guiana. Saving of civilisation from flood in a great boat has been described in south America also - Tales of Cochiti Indians - Bureau of American Ethnology. Bulletin 98 page 2-3. (4) Herodotus writes on basis of Egyptian calculations - (part 1 page 189) Seventeen thousand years (from the birth of Hercules) passed before the reign of Amasis. And even from Bacchus, youngest of the three, they count fifteen thousand years. 300 Siddhdnta Darpana Vayupurana tells 12 deva in 1st treta yuga. For further material - Bharata varsa ka Brhat Itihasa - by Bhagavaddatta, Pranava Prakasan, Delhi - 26 may be referred. E. Eras since Kali (i) King Yudhisthira ascended throne after Mahabharata war and the time since then is counted as Yudhisthira saka. • * Varahamihira writes in Brhatsamhita that according to old Garga, Saptarsi were in Magha during the reign of Yudhisthira. 36 years after that kali era started with death of Krsna • • * (2) Kali era - It started 36 years after Mahabharata war on the day Krsna died. After some months Yudhisthira relinquished his throne. According to Alberuni (part 3 p. 239), it started on 13th tithi of Asvina. There are thousands of documents mentioning this era. All text books count the day from kali beginning. Accordingly, it starts on 17/18-2-3102 B.C. Ujjain midnight on Friday. Its months are both Caitradi (luni solar) and mesadi (solar). Year expired in kali era is obtained by adding 3179 to saka year. (3) Saptarsi era - It was also called laukika kala and Sastra kala in Rajatarangini where it has been followed as the standard. This era began on Caitra sukla 1st tithi in kali year 27. Saptarsis remain for 100 years in one naksatra and after each century in a naksatra the years are counted afresh. Era is mentioned merely by the naksatra name in which saptarsi remain. Thus years are found by adding 46 to saka era, neglecting the centuries. (4) Old Saka era of Varahamihira - Brhat- samhita 13/3 tells that Saka starts 2526 years after Corrections to Moon 3G1 king Yudhisthira. This is 554 before Vikrama era. According to this saka era, time of Varahamihira was 427, i.e. 127 years before Vikrama era. (epoch of Pancasiddhantika) He himself writes in kutuhala manjari that he was in beginning of Vikrama samvat. Traditionally he is reputed to be one of 9 wise men with Vikramaditya who started Vikrama samvat. Such a great astronomer was needed to start the new era since Vikrama, Sri S.B. Dlksita also has stated in his history of Indian Astronomy vol II. p. 2 that the siddhantas mentioned in pancasiddhantika belong to 5th century before Saka era (new). Thus his epoch of 427 old saka is a convenient period about 100 years before Vikrama, not his time. The subject matter also is much older than Aryabhata during whose time the older, theories were extinct. (5) Sudraka or Sri Harsa samvat - (2644 Kali) Sudraka was also called Sri Harsa who was a king of Andhra Kula. Albiruni (chapter 49) has written that Sri Harsa was 400 years before Vikrama. Ain - Akbari (description of UjjainI) tells that difference between Aditya Ponwara (Sudraka) and Vikramaditya of Vikrama era was 422 years. Yalla in his Jyotisa Darpana (Saka new 1307) has written hi«»Pm U u I <^H1 (2345 or 2645) %#M<\: ^rf<m Taking 2645 as correct version, Sudraka era started in kali 2645 or 399 years before Vikrama. This Sudraka has written 'Mrcchakatikam' a famous drama. He ruled over Malwa, Kannauja, Kasmira etc. After 400 years, 2nd Vikram samvat became more popular and this era was forgotten. 302 Siddhanta Darpang This samvat was also called Krta samvat. King Samudragupta has written that (Krsna Carita) 'His rule was rule of law and religious, Hence his era was caled Krta samvat/ This was written as Malava samvat because of its start in Malava. Jain Acarya Hemacandra also has mentioned that rule of Sudraka was famous for righteousness (Kavyanusasana - Bombay edition p. 464). (6) Parad samvat - This is Indian name of Parthian era or Arsacid era starting in 246 B.C. in Iran. This was in use in West India. (7) Vikrama Samvat (kali 3044) - This was also called Sahasanka year. All Gupta kings used Vikram name, so this is connected with one of them. In old geneology, Samudragupta is con- sidered 93 years after Vikramaditya of Avanti (or Ujjain). In north India, it is Caitradi with purnimanta months. In Gujarat it is kartikadi and months are amanta. Jain inscriptions have called it Gupta era also. (8) Christian era started with British rule in India. (9) Salivahana Saka (78 A.D. - Alberuni has written (part-3) - One Saka king ruled in areas around Sindha river and through his tyranny he tried to destroy the Hindu culture. He was either a sudra of north west border or from a western foreign country. In the end, one king from east came and expelled him. After killing Saka king he was called Vikramaditya and another era started with him. m "A Corrections to Moon 303 This Vikramaditya might have been Skan- dagupta according to purana chronology. This is most popular, among astronomers. Amaraja Brahmagupta, Bhaskara II, have written that, 3179 years of kali had passed at the end of Saka king. Alberuni tells that Gupta - Ballabha samvat started 241 years after Saka. Gupta empire lasted for 242 years. Thus Gupta empire and Saka kala started together. Mesadi solar years are followed in Tamil and Bengal and caitradi lunar year is followed else- where. Lunar months are purnimanta in north India and Amanta in south India. (10) Kalchuri era or Cedi era : Kings of Bhoja kula ruled in Cedi (present Bundelkhanda) Yallaya in Jyotisa darpana has written that Bhojaraja samvat = Saka year + 50 According to this, it started in 28 AD or 85 years after Vikrama era. Keelhorn assumes it to start in 255 AD. considering Narasinha deo of Kalinga and of Dahal (M.P.) as same person. This year started from 'Asvina Sukla 1. (11) Valabhi era - One Vallabha ended the rule of last Gupta king who was a tyrant and started ths era in Saka 242. (This has already been mentioned in statement of Alberuni under para 9 above). Vallabhi king §uaditya had dispute with a merchant Ranka of his town. Ranka invited Hindu ^ng Hammira of Gajani (Afganistan). In a night 304 Siddhdnta Darpat raid Ballabhi was distroyed in Vikrama era 35 which was also referred. ■ m ■•.« (12) Hijri Era - This started with Islamic rule in India. (13) Kollam or Parasurama Era - This is knowri? as Kollam (western) Andu (year). This is used irl Kerala and in Tirunelveli district. This is sidereal solar year starting from solar month of Kanya in north Mahabar and simha month in south. This year runs in cycle of 1,000 years and present cycle is said to be fourth. It's 4th cycle started in Saka 747 or 824 A.D. According to Mahabharata, Parasuram was in sandhi of treta and dvapara (i.e. 24th part). Thus Parasurama must have been above 5000 years before 824 AD, may be 6000 completed years. (14) Nevara year started in 878 AD, with Karttikadi amanta months. It was used in Nepal upto 1768 AD. (15) Calukya Era - Calukya king Tribhuvana Malla started this era in 997 AD. (16) Simha Samvat - It was started in Gujrat in 1170 AD. Months are amanta and start with Asadha. • • (17) Bangali Fasall etc - Bangali san started in 593 AD. It is solar year and 1st month starting from mesa sarhkranti is called Vaisakha (it is called caitra elsewhere). All month names are lunar. Vilayati san started previous year with Kanya sankranti i.e. 7 months before Bangali san. The year is solar with lunar month names. This was Corrections to Moon 305 used in Orissa. Difference in rules of sankranti has already been explained. Amli Era also was used in Orissa with luni solar months. This year started from Bhadra sukla 12th (the month of kanya sankranti), which is supposed to be birthday of kings Indradyumna of Orissa, in purana era. Fasali san was started by Akbar. This started with same year number as Hizri era but it was solar calender to tally with harvesting time. In north India, it started in 1556 AD with Hijri year 963. In south India it started in 1636 AD when Hijri year had become 1046. Thus years in South India are 2 more. In north India, Fasali year started from Asvina Krsna 1 purnimanta. Then it was luni- solar. In Madras, it started with karka sankranti. The initial date was fixed by British on 13th July in 1800 AD and from 1st July in 1855 A.D. (18) Laksmanasena Era - It is current in MitMa region of north Bihar. This is Karttikadi, amanta and started in 1118 A.D. (19) Raja Saka - It started on Jyestha sukla 13th in Saka 1596 with coronation of Sivaji. (20) Ilahl Era - This was started by Akbar and also called Akbar san. It started on 14-2 - 1956 A.D. with his coronation. Its years and months are solar. Month names were Persian starting with Farvardin and each day of month had a separate name as in Persia. (11) Festivals and Yogas in India A. Rules Festivals are generally based on tithis except sankranti days. As a tithi generally covers a period of two days, a tithi may be counted on day when it is current on sunrise. But for religious purposes it may have to be celebrated on the previous day when it begins. Tithi for feast or fast 306 Siddhanta Darpt is observed on the day in which it covers the prescribed part. f For such purposes, a day is divided into 5't parts between sun rise and sun set - j (a) Pratah Kala - 6 ghatika from sunrise. J (b) Sarhjava - 6 to 12 ghatika from sunrise. J (c) Madhyahna - 12 to 18 ghatika from sunrise, j (d) Aparahna - 18 to 24 ghatika from sunrise. ; (e) Sayahna - 24 to 30 ghatikas from sunrise. \ Relevant parts of night are - (a) 4 ghatikas before sunrise are called arunodaya or usakala (b) 6 ghatika after sunset are called pradosa (c) 2 ghatika in middle of the night are called' j nisitha - midnight A tithi is purva viddha when it commences? more than 4 ghatikas before sunset of one day and; ends before sun set of the following day. A festival > on such a tithi is calebrated on the first day of| the tithi and not on the second. Tithi dvayam - when 2 tithis meet between? 18 and 24 ghatikas after sunrise, but a similar- meeting does not take place on next day. B. Festivals Connected with naksatras as well as Tithis In southern India, naksatras are often linked with solar months to observe a festival. Sravistha with Lunar sravana makes upakarma. Tithi festivals are also connected with solar months. When a sukla pakssa tithi falls twice in a solar month, the first is called a sunya tithi and only the second is celebrated. Corrections to Moon 307 (1) Pratipada (Tithi 1) Caitra sukla pratipada i.e. that which precedes the Mesa Sankranti, is the beginning of Hindu Lunar year. New year's day (Lunar) falls on the day when pratipada is current on sunrise. When there is an adhika caitra, that begins the year. This tithi is, therefore, called Vatsararambha. It is also Navaratrarambha. There is another navavatra starting on Asvina sukla pratipada. Karttika sukla 1 is Balipratipada or Balipuja and is purva viddha as to time. Bhadrapada bahula 1 is Mahalayarambha. Phalguna bahula 1 is Vasantotsava (2) Dvitiya (Tithi 2) Asadha sukla 2 is Rathayatra dvitiya or Rama rathotsava. Kartika sukla 2 is yama dvitiya or Bhratr dvitiya (sisters make presents to brothers in afternoon) Bahula dvitiya in 'Asadha, Sravana, Bhadrapada and Asvina is called Asunya sayana- vrata and fast is broken at moon rise. (3) Tritiya (Tithi III) Caitra sukla 3 is gauritritiya, also Matsya jayantl (afternoon), also Manvadi (forenoon). Vaisakha sukla 3 is kalpadi (forenoon), Treta yugadi (forenoon), Aksaya tritiya (special when combined with Wednesday and Rohini naksatra, forenoon), also Parasurama jayanti. Jyestha sukla 3 is Rambha tritiya, when BhavanI is worshipped at purva viddha. Sravana sukla 3 is madhu srava in Gujrat. ■4 308 Siddhanta Darpana >?& -..-■s Sravana bahula 3 is kajjali tritlya Bhadrapada sukla 3 is varaha-jayanti (after- noon); Haritalika, when Parvatf is worshipped, Manvadi (forenoon). It is also called siva tithi. Phalguna bahula 3 is kalpadi (forenoon) (4) Caturthi (Tithi 4) Sukla Caturthi in every month is called Ganesh caturthi on Vinayaka caturthi, the chief being Magna Caturthi (Ganesa jayantl). It is celebrated at midday. Tila caturthi is its another name; but is observed in evening. It i* also called kunda raturthi. Bhadrapada sukla caturthi is special when it falls on Sunday or tuesday. Similarly, bahula caturthi in every month is Sankasta caturthi and is a fast day for people in difficulties. Fast is broken at moon rise. If it falls on tuesday, it called Angaraka caturthi and continues till moon rise. Sravana bahula caturthi is the main Bahula caturthi, and cows are worshipped. (5) Pancami (Tithi 51 Caitra sukla 5 is Sri pancami. According t some, it is also kalpadi. Sravana sukla 5 is Naga pancami, when snak are worshipped. If the tithi starts within 6 gha after sunrise of one day and ends within 6 gh#! of sunrise on next day, the tithi is observed <* the first day. Bhadrapada sukla 5 is Rsi pancami. 308 Siddhanta Darpatu^ Sravana bahula 3 is kajjall tritlya Bhadrapada sukla 3 is varaha-jayanti (after-? noon); Haritalika, when Parvati is worshipped, Manvadi (forenoon). It is also called siva tithi. Phalguna bahula 3 is kalpadi (forenoon) (4) Caturthi (Tithi 4) Sukla Caturthi in every month is called Ganesh caturthi on Vinayaka caturthi, the chief being Magna Caturthi (Ganesa jayanti). It is celebrated at midday. Tila caturthi is its another name; but is observed in evening. It i« also called kunda taturthi. Bhadrapada sukla caturthi is special when it falls on sunday or tuesday. Similarly, bahula caturthi in every month is Sankasta caturthi and is a fast day for people in difficulties. Fast is broken at moon rise. If it falls on tuesday, it called Angaraka caturthi and continues till moon rise. Sravana bahula caturthi is the main Bahula caturthi, and cows are worshipped. (5) Pancami (Tithi 51 Caitra sukla 5 is Sri pancami. According to some, it is also kalpadi. Sravana sukla 5 is Naga pancami, when snakes are worshipped. If the tithi starts within 6 ghati after sunrise of one day and ends within 6 ghati of sunrise on next day, the tithi is observed, on the first day. Bhadrapada sukla 5 is Rsi pancami. 320 Siddhanta Darpan bhadra. A sukla saptami on a sankranti is calle< Mahajaya which is superior to eclipse for makirtj donations. Vaisakha sukla 7 is Ganga saptami or Gangot- patti (birih of Ganga - midday). Sravana bahula 7 is Sitala or Sitala vrata, tiirM purva viddha. Bhadrapada sukla 7 is called Aparajita Asvina sukla 7 - About this tithi Sarasvati isl worshipped under mula naksatra Karttika sukla 7 is kalpadi (forenoon) Margasira sukla is Surya vrata. Magna sukla 7 is Ratha saptami or Ma] saptami (time* arunodaya), Manvadi (forenoon) (8) Astami (Tithi 8) An astami, falling on Wednesday, is specia| and receives the name of Budhastami. The Sukla- ' astami in every month is sacred to Durga or Am piirna, Bahula-Astami in every month calle< Krsnastami, celebrated at purvaviddha, is sacred t< • • - * * Krsna. • • • Caitra sukla 8 in 'Bhavani utpatti'; whe] joined with Wednesday and punarvasu naksatra, bathing on this tithi is special. Sravana bahula 8 - JanmastamI, Krsnastami cd Krsna Jayanti (midnight) special when combine^ with Rohini naksatra; less so when joined or^ monday or Wednesday. Manvadi (afternoon). i Bhadrapada sukla 8 - Jyestha Gauri pujana vrata; when combined with Jyestha naksatra. Corrections to Moon 321 Bhadrapada bahula 8 - MahalaksmI vrata (purva viddha); Astaka sraddha. Asvina sukla 8 - Mahastaml, special when joined to tuesday. Karttika sukla 8 - Gopastami - worship of cows. Karttika bahula 8 - Krsnastaml, Kala bhairavastami or kala bhairava Jayanti. Margasira bahula 8 in Astaka sraddha in afternoon, the same is case with bahula 8 in Pausa, Magna or Phalgana. Pausa sukla 8 in special when on Wednesday with bharanl naksatra (Rohini or Ardra according to some). Magna sukla 8 is Bhlsmastami at midday. Magna bahula 8 is birth of Sita. (9) Navami (Tithi 9) Bhadrapada sukla 9 - Adukha navami. Asvina sukla 9 - Maha navami or Durga navami, Manvadi (forenoon) Karttika sukla 9 - Treta yugadi (forenoon) Margasirsa sukla 9 - Kalpadi (forenoon) Magha bahula 9 Ramadasa navami (10) Dasami (Tithi 10) Jyestha sukla 10 - Dasa-hara (destruction o\ 10 sins) Ganga- vatara. Asadha sukla 10 - Manvadi (forenoon) Asvina sukla 10 - Vijayadasami (afternoon special with sravana naksatra, Buddha Jayanti. 312 Siddhanta Darpanm (11) Ekadasi (Tithi 11) Every Ekadasi is sacred and has a separate name. It is called Vijaya when combined with Punarvasu naksatra. -.(. Month Sukla Bahula 1. Caitra Kamadi Varuthini 2. Vaisakha Mohiiu Apara 3. Jyestha Nirjala Yogini 4. Asadha • • Visnu sayanotsava Kamadi or * Sayani or Visnu Sayani Kamika (Visnu going to sleep) - 5. Sravana • Putradi Aja 6. Bhadrapada Visnu parivartanotsava or parivartini (Visnu Indira turning on his side) * - called Visnu Srhkhala • • • when 11th and 12th tithis meet in Sravana naksatra • 7. Asvina Papankusa or pasankusa Rama 8. Karttika Prabodhini (Awakening of Visnu), Bhisma pancaka Vrata commences Utpatti 9. Margasira Moksada Saphala 10. Pausa • Putrada or Vaikuntha * • sat-tila • ■ ekadasi, Manvadi (forenoon) 11. Magha Jaya Vijaya 12. Phalguna Amalki Papa mocini 12. Dvadasi (Tithi 12) This is called Mahadvadasi in the following circumstances - 11th tithi current at sunrise on two . successive days : the next dvadasi is called Unmflani. Corrections to Moon 313 12th tithi current at sunrise on two successive days - first dvadasi is called Vanjull. 12th tithi followed by a full moon or a new moon tithi, current at two sunrises - Paksa Vardhini. 12th tithi with Pusya naksatra - Jaya - do - Sravana naksatra- Vijaya - do - Punarvasu naksatra - Jayanti - do - RohinI naksastra - PapanasinI ~ Vaisakha sukla 12, with Hasta naksastra, guru and mangala in simha, surya in mesa - Vyatlpata Asadha sukla 12, commencement of caturmasya vrata Sravana sukla 12, Visnoh pavitraropanam Bhadrapada sukla 12 - Vamana Jayanti (mid day); called Sravana divadasi when with sravana naksatra, specially on Wednesday. Asvina bahula 12 - Govatsa dvadasi (evening) Karttika sukla 12 - (i) End of caturmasya vrata which began on same tithi in Asadha. (ii) Prabodhotsava or Utthana dvadasi (preparation for waking Visnu) (iii) TulasI vivaha (Marriage of Visnu with TulasI plant) (iv) Manvadi (forenoon) Magna sukla 12 - Bhlsma dvadasi Magna bahula 12 - Tila dvadasi or vijaya when with Sravana naksatra (when previous Magna is adhika). 314 Siddhanta Darpanm 13. Trayodasi (Tithi 13) Caitra sukla 13 - Madana trayodasi or Ananga pujana Vrata (purva viddha) Bhadrapada bahula 13 - Kali yugadi (after- noon) (ii) Magha trayodasi - when with magna naksatra (iii) Gaja chaya .- when with magha naksatra and sun in hasta. Asvina bahula 13 - Dhana trayodasi Magha sukla 13 - Kalpadi (forenoon) Phalguna bahula 13 (i) Varum when joined with Satabhisaj (ii) Mahavarum - do - + Saturday (iii) Maha-maha -varum - when joined with Satabhisaja naksatra + Saturday + subha yoga (14) Caturdasi (Tithi 14) Bahula Caturdasi in every month is Sivaratri Vaisakha sukla 14 - Narasirhha Jayanti (sunset) : special when joined with svati naksatra + Saturday Sravana sukla 14 - Varahalaksmi Vrata Bhadrapada sukla 14 - Ananta caturdasi 3 Asvina bahula 14 - Naraka caturdasi (moon rise), Dipavali may fall on this tithi if with svatf naksatra (normally on Asvina bahula 15) Karttika sukla 14 - Vaikuntha caturdasi (mid night) Margasirsa sukla 14 - Pasana Caturdasi Magha bahula 14 - Maha Sivaratri (mid night when Sravana naksatra is current). Special when combined with Sunday or tuesday and siva yoga. 315 Corrections to Moon 15. Sukla PancadasI (Tithi 15) or Purnima A sukla 15 or purnima is called somavati when it falls on monday and is special for donations It is called cudamani - when further joined with a lunar eclipse. Special names are given below- Caitra Purnima (1) Manvadi (forenoon) (2) Hanumana Jayanti (3) Special for bathing when combined with sunday, thursday or Saturday. Vaisakha purnima - Kurma Jayanti (late after noon) Jyestha purnima (i) Manvadi (fore noon) (ii) Vata purnima or Vata savitri (purva viddha) (iii) Maha Jyestha when moon and jupiter are in Jyestha naksatra and sun in Rohinl Asadha Purnima (i) Manvadi (forenoon) (ii) Siva sayanotsava or Kokila vrata or Vyasa puja Sravana purnima (i) Rk yajuh Sravani (for followers of Rk and yajurveda (ii) Raksa bandhana (tying a string round the arm) or Rakhi purnima or narali purnima (throwing coconuts into the sea) (iii) Hayagrlva Jayanti Bhadrapada purnima - (i) Kojagari purnima or kojagara vrata (mid night) Laxmi and Indra worshipped; games of chance. (ii) Navanna purnima - when new grain is cooked. 326 Siddhanta Darpana Karttika purnima (i) Manvadi (forenoon) (ii) Caturmasya vrata ends (iii) Tripurl purnima or tripurotsava (iv) Special when joined with Krttika naksatra (v) Maha Karttiki, when joined to naksatra Rohini or when moon and jupiter both are in Krttika naksatra. (vi) Padmaka yoga when moon in Krttika and sun in visakha Margasirsa purnima (i) Dattatreya or Datta Jayanti (evening) (ii) Special for donation of salt when joined with Mrgasira Magna purnima - maghi - when moon and jupiter both are in Magna naksatra Phalguna purnima (i) Manvadi (forenoon) (ii) Holika or Hutasani purnima (evening) (16) Bahula Pancadasi (Tithi 15) - Amavasya A solar eclipse on Sunday is cudamani and is special for donations. Sravana amavasya (at beginning of next month bhadrapada as in all cases) - Pithori or Kusotpatini Bhadrapada amavasya - Sarvapitri or Mahalaya amavasya, special when sun and moon both are in hasta Asvina amavasya - DTpavali, with previous or following tithis; that on svati naksatra is special. Pausa amavasya - (i) Ardhodaya when joined with Sunday in day time + Sravana naksatra + Vyatipata yoga. (This can happen only when some previous month is adhika) Corrections to Moon 327 (ii) Mahodaya - when any one of these special features is lacking. Magna amavasya (i) Dvapara yugadi (after- noon) (ii) special for sraddha when joined with Satabhisaja or dhanistha naksatra. Phalguna amavasya - Manvadi (afternoon) Notes - (i) Many festivals differ due to interpratation by different sects or regions. (ii) The list is not exhaustive (iii) There were 14 manus (7 yet to come). So 14 days are manvadi. (iv) Birthdays also differ according to inter- pretation whether original reckoning was solar or lunar. Birthdays of other sains like Tulasidasa, Nanaka, Kabira or Raidasa etc are also calebrated. (v) There are many other local festivals. C. Methods for Citation : (i) Normally the gata or expired years, amanta months and Caitradi years are given. A lunar year begins only when the solar year begins with mesa sankranti. Thus it is same as counting mesadi solar years. Compared to this, the current years in christian era are counted. (ii) Sometimes Varttamana or current years, purnimanta months or karttikadi lunar years are also given, peculiar to a system of samvat. (iii) An era is called year, samvat, samvatsara, san (arabic or persian), saka etc. Thus saka is not only saka era but year in any era is called saka. (iv) Samvatsara is named in 60 year cycle of guru varsa or Jovian years. In south India it is merely a solar year with Jovian name. In north 318 Siddhanta Darpm India it is the Jovian year actually completed a| the beginnig of a solar year or year at the moment. Jovian years are also named in 12 years cycle, when it completes one revolution. The current rasi of Jupiter is name of year. Alternatively, Jupiter! years are named on lunar months, corresponding to solar rasis in which Jupiter is present. Maha is abided before these lunar months to indicate that they are years (Jovian) References : (i) Spherical astronomy by W.M. Smart, Longman Green, London or by R.V. Vaidya, Payal prakasana, Nagpur etc can be referred. Godfray has written a book 'Moon' only about Kinematic theory and perturbations of moon. (2) For finding correct positions, a Nautical Almanc can be referred. (3) History of calenders can be referred to the relevant article in Encyclopadia Britanica by Fotherington. History of calendar has also been published by govt, of India, publications division, being part C of report by National commission on calendars under Dr. M.N. Saha. (4) History of Astronomy by S.B. Diksita, Govt, of India, or by S.N. Sen and A.V. Sub- barayappa, published by Indian National Science Academy, Delhi - 2. (5) Bharata Varsa Ka Brhat Itihasa by Bhagavaddatta, Pranava Prakasana, Delhi - 26. (6) Indian chronology by L.D. Swami Kannu Pillai. (7) Reference for deciding festivals is Nirnaya Sindhu by Kamalakara Bhatta. Corrections to Moon 31 $ Translation of Text (Chapter 6) Verses 1-3 - Scope - Now I (author) write accurate panjika for getting quick results in marriage, sacred thread ceremony, house construc- tion, yajna and birth caremonies etc. With help of this, accurate position of sun and moon are known and kranti, sara, lunar and solar eclipse, conjunc- tion of planet and naksatra, rising and setting, mahapata, tithi, naksatra, yoga and karana etc can be calculated. Old astronomers assumed maximum increase of 5 danda and decrease of 6 danda in a normal tithi of 6 danda. Due to this their panjika was inaccurate, because actual increase or decrease limit of a tithi is much more. After describing rough panjika according to old school in last chapter, now I an explaining the method for accurate panjika. Calculation as per these rules will give correct time for auspicious works and the tithis etc can be actually seen. When direct observation proves the accuracy, no further logic is necessary in support of these rules. Mean sun and moon corrected only by mandaphala give correct position at amavasya and purnima. This has been described in previous chapter. Verses 4-6 - Need for further corrections Panjika has five limbs - vara, tithi, naksatra, yoga and karana - so it is called pancanga. Except the first part vara, all others depends on sun and moon. Hence these will be accurate if sun and moon are accurate. Manda paridhi of moon and sun is taken same for rough and accurate methods 320 Siddhanta Darpana both. Hence sun and moon corrected by man- daphala is called 1st (corrected) planet. From this 1st graha and 1st ravi sphuta gati, we calculate diameter of planet, time upto parva sandhi (i.e. purnima or amavasya), bhuja, mandakarna and lambana correction of sun. * Motion of moon is very complicated. After long period of observation, I have thought it necessary to have 3 more correction in addition to mandaphala. These four corrections are - Manda, Tungantara, Paksika and digamsa. Verses 7-9 - Tungantara correction Tungantara kendra = Candra Mandocca - (sphuta ravi + 3 rasi) - in Suklapaksa or = Candra mandocca - (sphuta ravi - 3 rasi)! - in Krsna paksa Find out bhuja jya of tungantara kendra. j Tungantara bhujaphala (its bhuja jya) isj multiplied by 16 and divided by radius (3438). Then| it is multiplied by bhuja jya of difference of sphutaf ravi and sphuta candra and again divided byj radius. 1 "4 Result in kala etc. is multiplied by 1st candraj sphuta gati and divided by madhya candra gati| (790'i35"). Result in kala etc will be tungantara| phala. When tungantara kendra is 0° to 180°, thte| is added to 1st sphuta candra otherwise substracted.| We get second sphuta candra. 1 Verses 10-12 - Paksika phala J Paksika Kendra = 2nd sphuta candra - sphut^ ravi. Lapsed and remaining parts of the kendra its quadrant are found. Lesser of the two Corrections to Moon 321 converted to kala and multiplied by 2. Jya of the resulting angle is divided by the hara, to be calculated as below, gives paksika phala in kala etc. When candra (2nd) is in 1st half of paksa, paksika phala is added to 2nd candra, otherwise it is deducted. Paksika hara is found by substracting 1st sphuta sun separately from mandocca of surya and candra. Bhujajya of the two remainders is calculated and multiplied together. Product is divided by 180 and to the quotient we add 90. Result is the hara for paksika phala. Verse 13 Digamsa phala * Mandaphala calculated from sphuta sun is divided by 10; multiplied by sphuta candragati, and divided by madhya candra gati. Result is digamsa phala. The mandaphala being positive or negative, digamsa phala is added or substracted from 3rd sphuta candra. We get 4th sphuta candra which will be accurate position. Verses 14-15 : Reasons for correction - In plane surface snake moves in a wave like motion, but at the time of entering a hole, its motion becomes straight. Similarly, moon deviates from the mandocca resultant motion normally, but on purnima and amavasya, these deviations vanish and only mandocca effect remains. When snake enters a hole, its wavy motion ceases under pressure from narrow sides, but its natural forward motion also is affected. Similarly on parvasandhi, moon is not affected by tungantara and paksika sanskara, but digamsa phala is still effective (in addition to mandaphala). % 322 Siddhdnta Darparui Comments : Fortnightly venations in orbit due to sun effect are not evident on parvasariclhi (purnima or amavasya) because sun, moon and earth are in a straight line. Howevere, total attraction of sun, varies according to its own variation in distance, causing minor corrections of digamsa phala. Parvasandhis are like a hole for snake, hence the simili. Verse 16 - Rough and accurate correction for sun - For correcting accurate sphuta of ravi, accurate chart should be used and for rough sphuta, rough chart is used. Use in reverse order will create errors. Verses 17-21 - Accurate sphuta gati of candra - .Accurate tungantara phala is multiplied by radius (3438) and divided by Jya of difference between 1st sphuta candra and accurate sphuta ravi. Result is multiplied by kotijya of difference of (1st sphuta ravi - candra) and divided by radius (3438). Result added to 1st sphuta candra gati phala gives 2nd gati phala. This is added when rnanda kendra is 90° to 270°, otherwise substr acted. Paksika phala in kala etc. is squared arid deducted from the square of maximum piksifca phala. Square root of remainder is multiplied by difference of 2nd candragati and sphuta sufya gala and divided by half radius (1719). Result is adde«f to 2nd gati when candra is in 1st half of Sukia paksa or 2nd half of krsna paksa. Otherwise, it is deducted. This is third sphuta gati of moon. This *third gati will be accurate. Difference of sphuta candra on two successive days also gives suffiderit- Corrections to Moon 323 ly accurate gati for calculation of tithi and auspi- cious works. Comments : Corrections have already been explained in the introduction of this chapter. This explains the change of speed due to two paksika variations due to sun's attraction. Last correction is due to change in distance of sun which is negligible within a day and correction is not needed for r already small effect. Tungantara phala = - 160' cos (0—a) Sin (D-0) where D = moon corrected for mandaphala, 6- , Longitude of sun, a = mandocca of moon. For a short period, only D is variable which is position of moon. Hence speed due to this correction is by differentieting Sin (D-0) = - 160' cos (0-a) Cos (D-<9), d(D~0) RxTungantara phala R cos (D - 0) = Rsin(D-0) ' R Thus we get the formula Paksika phala = 38'12" Sin 2 (D-0), 38'12" = max phala = P where D' = 2nd sphuta moon. Gati due to paksika phala is its differntial = 38'12" cos 2 (D-6) x 2 d(D-0) = 38'12" / 1 _ sin 2 2 (D _0 } x 2 (dD - dS) = J Vp 2 - (Paksika phala) 2 x ( 2n d candra gati - ravigati) x 2 324 Siddhanta Darpana r —— 2nd candra gati - Ravi gati = V P 2 - Paksika Phala) 2 x wi This is the formula given above Verse 22 : Sthula value is not entirely useless, it is good for daily use. But adverse moments like { visti, to be strictly avoided, should be calculated j only through accurate motion. Verse 23 : Phases of moon Sun deducted from moon gives the kendra, when this kendra is in ist 6 rasi i.e. 0° to 180°, it is sukla paksa. When moon is ahead of ravi by 180° to 360° it is krsna paksa.First 3 rasis are 1st half of suklapaksa, then upto 6 rasis it is 2nd half of sukla paksa. Similarly in krsna paksa, Ist half is from 6 to 9 rasis and 2nd half is from 9 to 12 rasis. One fourth of every paksa (i.e. 45° difference) is called paksa pada (quarter). Verses 24-26 - More correct motion of moon - Now I am telling more accurate motion of moon which needs to be calculated for eclipse. For this gati, we find gata and gamya kala (lapsed and remaining) times of parvanta (purnima or amavasya). From that, true moon is found out. From this sphuta gati, sparsa, moksa, sthiti etc periods of eclipse are accurately known. First gatiphala is kept in two places. At one place it is multiplied by parama tungantara phala (160) and divided by parama mandaphala (300'50")/ At second place it is multiplied by Ist sphuta gati and divided by madhyagati (790'35"). Result of both places are added. The sum is added to madhyama gati of candra when manda kendra is Corrections to Moon 325 in 6 rasis starting from karka (90° -270°), otherwise it is deducted. This is true mean speed of moon. This is again kept at two places. At one place, sphuta surya gati is substracted and divided by half of hara. Here Hara = bhuja of (candra mandocca - suksma surya) x Bhuja of (ravi man- docca - suksma ravi ^180° +90°. Quotient is added to the true mean motion of moon at second place. Sum is true motion of moon. Kotiphala x Kendragati Comments - 1st gatiphala = — : — r .cos m x 6m = _ Thus the above formula is <5 m 160 r cos m — — - r cos m ( + 1 + — - ) R v r R. Gatiphala of Tungantara is 160 cos m. cos (D-0) m = D - a = mandakendra, cos (D-#) is 1 for parvanta as D-^ = 0° or 180° d M Hence, tungantara gati phala = 160 cos — r— This is the first term of above formula • d M 160 160 cos m s — — - . r cos m . = — dm R r R Remaining terms are second order corrections in mandaphala itself. Second step of correction amounts to paksika correction as stated earlier after verse 21. 326 Siddhanta Darpana Verse 27 - Accurate panjika - Use of one rough panjika for normal works and another accurate panjika for important works will not be appreciated by anybody. Hence only accurate panjika should be used, even though it involves hard labour. It alway deserves more respect. Verse 28 - Definition' of true planet * We are on the surface of earth. The point of sky where line from earth's interior centre to our location point on surface meets is called 'svastika' (vertically upward point). When the planet is seen on the great circle from kadamba (pole of ecliptic) through svastika and calculated position of graha on ecliptic, graha is called spasta. Note - This will be explained fully in chapter 7 and lambana samskara for solar eclipse. (Chapter 9) Verse 29-31 : Need for calculating true planet - According to old teachers, all auspicious works are done only according to this true graha. For this purpose bhagana and bija corrections are done to the planet. For daily and special works of vaidika and smartta type, true position of all planets are needed. But correction to moon is needed more, because pancanga is based on moon's position. Hence, accurate corrections like tungantara, paksika and digamsa etc. have been thought of. Even after these three sanskaras, there is difference of 2-3 palas (upto 1 minute) in the calculated and observed position of moon. But it is preferable to error of upto 14 ghati (about 6 hours) which will occur without these corrections. 327 Corrections to Moon 0£ -' Only Brahma can know how to eliminate this small error. Note : Though these corrections are great improvement, some error will always remain. Error withhin 1 minute is sufficient for day to clay work. More accurate position is needed for scientific works. Every formula will give some error, though it is about 1/10 seconds or less in modern methods. Verse 32-33 : Lambana and sara A planet will be seen in different position, when seen from earth's centre (which is calculated) and when seen from surface (where we are located). The angular difference between two position is called lambana. (In vertical position, it is already in line from earth centre to surface, hence lambana will be nil). Distance of the planet from kranti vrtta along great circle from pole of ecliptic (kadamba) to centre of planetary disc (also passing through svastika - vertical up point) is called sara or viksepa. Calculation of lambana and sara is called drk-karma (change of axis). This is needed only for lunar and solar eclipse and conjunction of planets. In that context only, it will be calculated. It is not needed for calculation of tithi and naksatra etc. Verses 34-39 : Authorities on need of true planets - Vrhatsamhita (Varahamihira) has stated - If grahana occurs before calculated time, then damage to foetus or child in womb, or war with weapons occurs. If it occurs after calculated time, 32 ° Siddkanta Darpana then damage to crops, loss of flowers and fruits and fear for people occurs. Garga sarhhita states - The result of having eclipse before or after calculated time has already been stated. Persons knowing true planets, never have error in timing. If every (astronomical) event occurs according to calculated time, then enemies of kings are destroyed and troubles cease. People become happy, being free from fear and disease. Vasistha states (not known in which text) - Tithi etc. should be decided according to that theory only which gives true position of planets. Sakalya samhita states - Corrections to calcu- lated position of planets should be done after observing them through instruments like nalika (tube or telescope) gola (mirrors or sphere), turiya (Fourth - compound telescope). Correction of observed error is called bija sanskara. After that correction only, all rules will be correct. Result of direct observation (pratyaksa) cannot be ignored. What is use of the gold ornament which cuts the ears ? Similarly what is the use of that sastra whose results are not actually seen ? Verses 40-42 : No need of lambana for tithi cal- culation The people who talk of lambana for calculation of tithi etc, do not know its meaning. Explaining them is like talking to a deaf. Spasta graha is known from the point of intersection of line from earth centre to the planet, when the graha is seen at that place, it is called spasta (or true) planet. Corrections to Moon 329 If graha is seen from earth's surface, tithi will be different for different places due to separate larrtbana corrections, hence tithi needs to be calculated from earth centre, so that it is same all ove the world. Verses 43-46 : Bija sanakara - When graha is not same as per ganita (calculation) and drk (observatoin), it cannot be used for auspicious works. So I describe the bija karma, i.e. corrections to calculated position to tally it with observed position. Bhaskracarya (in BIjopanayana) has stated - After daily observations of moon, I have observed that moon is seen 112' lipta east or west from its calculated position. These are the minimum or maximum values of Bija. In Surya siddhanta - Sun himself has stated in the end that he was explaining bija for good of the world even though it was a secret; after praying to gods and vedas. In Brahma-sphuta siddhanta - Graha ganita (calculation from planetary theory) as told by Brahma himself was lost (became erroneous) after lapse of long time. So Brahma gupta, son of Jisnu, seeks to correct it with bija-sanskara. Verse 47-52 : Origin of Tungantara correction - The error in calculated position of moon upto 112' Hpta (stated by Bhaskara) is probably due to distance of moon from ecliptic, so it should be related to the maximum viksepa (281' lipta). Because, after adding 3 rasis of sayana moon, Jya °f its kranti is 1370'. This multiplied by maximum 330 . Siddhanta Darpana viksepa and divided by radius (3438) gives 112 kala. This is the same amount which is found by calculating difference between calculated planet and observed planet. Bhaskarcayra has called it Ayana dik—karma, there are different types of Ayana karma in other siddhantas. So it should be called a bija sanskara. I have called it parama tungantara phala. It appears from drk-karma of Bhaskaracarya that maximum value of tungantara sanskara is continuing since long ago. According to ancient teachers, it fluctuates, so they have advised to correct moon with bija sanskara. To know the change in maximum value of tungantara phala, moon will be corrected after one thousand years. The error from true moon will give the value of change. Notes : Candrasekhara has not understood the theory or reasons behind this tungantara correction, But from the nature of variations, he has correctly assumed the position of maximum deviation and hence has got the correct formula. Verses 53-57 - Variations in duration of tithi - A tithi which includes aparahna of two consecutive days has beeen called suksma tithi in smrtis. Thus such a tithi has more than 66 dandas * • ■ (as aparahna period is 6 danda and a day is of 60 dandas). According to smrti, if tithi just touches one evening and is over before first half of day, then the sraddha of that day should be done on next day. It should be over by 'kutupa' of next day. Corrections to Moon 331 (Here 'Kutupa' means 8th muhurtta of the day time out of 15 muhurtta = 30 dandas between sunrise to sunset). Thus tithi = sayahna 6 danda + night 30 danda + half day time 15 danda = 51 danda. Gautama smrti has stated - If in sayahna (evening) of caturdasi (Krsna paksa), amavasya starts and is over before midday, then sraddha should begin in kutupa muhurtta (14 to 16 dandas from sunrise) and should be over by rohana muhurtta (12-14 danda after sunrise) on next day. This is called amavasya sraddha. In sukla and krsna paksa, on 7th, 8th and 9th - the three middle tithis, maximum difference in tithi duration is less than 6 danda. So these tithis have only 5 types of classifications - the 6th category of above six danda difference from 60 danda doesn't exist. Other 12 tithis in both paksa have 6 types 6f class. If smrtis are interpreted in this manner, there is no error in drk siddha (true) calculations. Ancient teachers, didn't observe the daily location of moon in constellations. With rough calculation, there is variations in middle tithis also upto 14 dandas (i.e. I 7 danda from average). But they had strived for accuracy, only at the end of a paksa when wrong eclipse time will cause insult to the astronomer. Notes - Traditional view about variation of tithi is 'Bana vrddhi, rasaksayah' i.e. increase upto 5 dandas and decrease upto 6 danda. This gives tithi limit from 54 to 65 dandas. But Candrasekhara 332 Siddhdnta Darpana has found corroboration from smrtis that it is actually from 51 to over 66* dandas. * Verses 58-67 : Suksma naksatra of unequal divisions - Now I explain the method to calculate suksma naksatra (unqeual divisions) for use in journey, marriage and sacred thread ceremoney etc as decided by sages like Garga, Vasistha. Mean motion of moon in a day (790'35") is the extent of suksma naksatra. One and half times this value is the extent of these six naksatras equal to (1185'52"18"') - (4) RohinI (7) Punarvasu (16) Anuradha and three Uttara naksatras (12) Uttara phalgunl (21) Uttarasadha (26) Uttara Bhadrapada. Half extent (395'17"26"') is of the six naksatras (9) Asiesa (15) Svati (18) J^estha and (24) Satabhisa Remaining 15 naksatras have unit extent (790'35"). Deducting the total of these 27 naksatras (21345'41"5'") from kalas of full circle (21,600), remainder (254'18"35 ,,/ ) is the extent of Abhijita which comes between (21) Uttarasadha and (22) Sravana. We substract the kala of as many naksatras from sphuta graha as it is possible. It is the number of completed naksatras. Remainder (gata) kala of the graha is the lapsed part of current naksatra. This part deducted from full extent of current naksatra gives remaining part (gamy a or bhogya kala). Gata and bhogya kala, separately multiplied by 60 and divided by sphuta gati of graha, give the lapsed or remaining time of the graha in current naksatra. Each of the 28 naksatras of unit, half, one half length or Abhijit being divided by 4 gives its one pada (quarter). <>i Corrections to Moon As per rough rule, rasi of 1800 kala contains 9 naksatra pada (27 naksatras excluding Abhijit have 27 X 4 = 108 pada = 12 rasi X 9). Thus 108 pada in 12 rasis are according to mean equal values of naksatras. With suksma rule, 1 rasi doesn't have complete number of naksatra padas. For example, at the end of 4 rasis 4X9 = 36 naksatra pada or 36 -s- 4 = 9 naksatra till Aslesa will be completed and 5th rasi sirhha should start with 10th naksatra. But according to suksma calculation magna naksatra starts 8° before simha rasi itself. Notes : (1) There are three measures of a pancanga for approximately one civil day. Week days are for fixing current routine of work and a day more than seven days ago or in future is not referred by the week day. Thus in modern university history books, even for modern eras, week days do not figure. This doesn't mean that use of week days is not common in modern days. Due to temporary nature of weekdays, and use for astrology only, they have not mentioned in histories of Ramayana and Mahabharata and in vedas. This should not mean that week days were not known in ancient India, as it is concluded by so called modern scholars. Another weakness of a week day is that it starts from local sunrise time in Indian system (local midnight in Gregorian or christian calender, local evening in Jewish and Islamic calender). Thus in all systems, it starts at different time at different places. Thus it cannot be made a world rference. (2) Technically, tithi starts on same time all over the world. But for civil purposes, only the tithi current at sunrise is counted, hence it may SS4 SiddhdntQ Darpatyi ■ i be useful for religious functions, but civil tithi willj be different in different places. Another defect is! that it is a mathematical calculation, even when! moon has risen, only the approximate tithi can be i known from its phase by rough eye estimate. However, naksatra can be measured more accurate- ly even with seeing moon's position among stars; by naked eye. Even for calculation purpose, it! doesn't suffer the errors in finding true position^ of moon, as it can be seen by direct observation,- This is the reason that all the important events in Mahabharata, Ramayana and Pur ana are indicated by naksatra of moon (instead of tithi) in addition of the lunar month. This is evident to the whole public and easily identifiable time in distant future.; (3) When naksatra extent is made exactly equalf to the mean motion of moon in one day, some part of the full circle will be left out as the moonj takes more than 27 days for a siderial revolitions. \ Thus 27 naksatra equivalent to 27 days motion of moon, doesn't cover the circle completely and a small 28th naksatra abhijit is introduced equivalent to extra time beyond 27 days taken in moon's sidereal revolution. Reason of unequal division is that, along the path of moon in sky, inclined at 5° angle with path of ecliptic, sufficient bright stars are notj available for all naksatra divisions. The three vacant^ places were identified with their preceding stari groups causing division of 3 naksatras in purva^ and uttara parts - Phalguni, asadha and bhadrapada (or prostha pada - old name). These three vacant places and 3 other star groups having lesser gaps - were given 1-1/2 times the length. Correspond- 1 Corrections to Moon 335 ingly, the length of 6 naksatras was reduce to half to compenisate the excess. (4) Visvamitra made equal divisions for each of 27 naksatras and further divided them into 124 parts each for accurate calculations of solar and lunar naksatras at the end of day, paksa or half year. Thus he created a different naksatra system - which is proverbial creation of stars by him. This has been explained in introduction, while explain- ing vedanga jyotisa. Corresponding to 24 original naksatras, there are 24 letters in a Gayatri chanda. But with 3 extra naksatras by division of 3 into purva uttara parts, 3 extra letters (vyahrtis) were added to Gayatri mantra whose sage is Visvamitra, making 27 letters in it. This corroborates the view that number of verses in Rk veda and number of letter in its chandas are based on astronomical measurements at regular intervals. This unequal division will be more clear when longitudes of identifying stars are discussed. Verses 68 - 71 - Sankranti i.e. crossing from 1 division to another Exact point of sankranti of a rasi is when centre point of a planet's disc reaches the last point of the rasi. This suksma sankramana is known by name of rasi which is to be entered, not the past rasi. Complete sankranti period is the time taken by complete disc of graha from touching the border point to its complete crossing. To find the sankramana kala, drameter of graha birhba (disc) in vikala is divided by graha gati in kala. Sankramana kala is obtained in danda etc. Within 336 Siddhanta Darpar ■!? sankranti period, surya gives very favourable! results. Planets give mixed results of both rasis during sahkranti period. Similarly while crossing over from one naksatra to the next, as long as the border point is covered by birhba (disc) of the planet, it gives results of both the naksatras. Candra birnba (disc) in Vikala, separately being divided by (1) difference of ravi and candra gati (2) candragati, (3) sum of candra and ravi gati, gives respectively the sandhi time of (i) tithi or karana (2) naksatra and (3) yoga. Verses 72-74 - Different circles - Due to effects of ucca, kranti and pata, many circular orbits are formed. Orbit due to attraction of slghra and manda ucca is called pratimandala (eccentric circle - explained in previous chapter). Path of kranti is called apamandala (to be explained in Triprasnadhikara).' Due to deviation of graha from kranti vrtta due to pata, another circle vimandala is formed which is path of pata (apamandala and vimandala are great circles perpendicular to ecliptic and will be explaiend in next chapter. Verses 75-91 : Precession of ecliptic and Ayanamsa Point of . intersection of kranti vrtta (ecliptic plane of sun's orbit) and visuva vrtta is. called pata which moves in the opposite direction to the normal motion of planets. Completed revolutions (bhagana) of pata in a kalpa are (6,40,170) as observed by the author. Corrections to Moon 337 This pata is above all planets and circle of naksatras (slowest rotation indicates farthest dis- tance). This moves the naksatra vrtta from east to west in plane of kranti vrtta. When this pata is in six rasis beginning with mesa (0° to 180°), it takes the naksatra and planets etc 27° towards east. When it is in six rasis starting from tula (180° to 360°), it takes the naksatras etc 27° towards west. Due to this pata, planets like ravi and naksatras starting with Asvini are seen towards north or south from ecliptic even on the position of 0° kranti. To find out kranti pata for desired day, kalpa revolutions of kranti pata are multiplied by ahargana and divided by savana dinas in a kalpa (i.e. 15,77, 91, 78, 28, 000). We get the complete revolution numbers and from remainder rasi etc. of kranti pata. The result in rasi etc is substracted from 12 rasi and remainder is converted to bhuja according to quadrant and then to kala. Bhuja kali divided by 200 is called calansa. This is also called ayanamsa. Raminder after division by 200, is multiplied by 60 and divided by 200. We get kala of ayanamsa. Motion of ayanamsa in one day is 9/28 para etc. At the beginning of karanaba (1869 AD, mesa sankranti at Lanka), ayanamsa was 22°1'51"45'"42"" etc. When krantipata is in six rasis beginning with tula, then ayanamsa is negative and, when in six rasis beginning with mesa it is positive. 338 Siddhdnta Darpana According to Surya siddhanta - Ayanamsa corrected graha (or say ana graha) only is used for calculating kranti, chaya, carkhanda etc. Motion of kranti pata can be seen at the time of visuva sankranti (sayana karka sankranti in uttarayana and makara sankranti in daksina ayana). Saptarsi, Agastya and Yama and the stars close to them have no motion due to kranti pata (They are near north or south pole and very far from ecliptic). Their motion in naksatra mandala towards east indicates that naksatra circle has moved west wards. Seeing west ward motion of Saptarsi etc means that naksatra circle has moved eastwards. With this concept, astronomers calculate the sara of naksara, which is north or south deviation from kranti vrtta along circle perpan- dicular to it. Position of sun calculated from shadow (chaya) is different from mathematical position of true sun. This difference is ayanamsa. This ayanamsa is also moving eastwards. If calculated true sun is more than sun found from shadow, then ayanamsa is moving west wards. At the time of karka and makara sankranti, when kranti of sun is equal, the rasi etc of sun at both points is added, ei Their half is ayanamsa. E2 / When sayana karka or£ makara sankranti is seen before nirayana sankrantis then ayanamsa will be added, otherwise it will be deducted. Figure 2 Corrections to Moon 339 Note - (1) Newton's explaination : (Figure 2) C is pole of ecliptic EL'L. Let Ti be mid point of E and L and thus the first point of mesa for year 1 . Then the celestial pole is at Pi and celestial equator is Ei,TiQi Due to precession of equinoxes, the first point of mesa is slowly moving in backward direction LT 2 E along the ecliptic. If Ti shifts to T 2 in year 2, the celestial pole shifts to P 2 along a small circle P1P2P3 where CP is obliquity of the ecliptic. The celestial equator assumes a new position E2 T2 Q2 in year 2. The celestial pole Pi goes round the pole of ecliptic C and it makes a complete circle in a period of about 26000 years. R Figure 3 In Fig 3 - if earth is homogenous sphere, the force of attraction of sun will act, as if the mass is concentrated on its centre C, But it is an oblate spheriod, whose polar axis is shorter than equatorial axis by 43 Kms. The main pull due to sun is still along CS which keeps earth in orbit round the sun. But the bulge at equator EEi suffers additional pull. The nearer portion of bulge at Ei 1S attracted more and E less. This extra forces at Ei and E are equal and opposite in the direction °f sun, but line EEi is inclined at an angle with C S. Hence it is a couple which tries to bring earth's -■A'* 340 Siddhdnta Darpana ■} equator in plane of ecliptic. Due to this couple, j precessional motion arises. Overall reason of precession of solar orbits is that each planet influences the other and net effect : is to bring angular momentum vector of all planets nearer to the direction of total angular momentum of the solar system. This mutual perturbation has a cycle of around 28,000 years. Due to motion of sun round the galactic centre also the angular momentum vector of solar system is turning in direction of galaxy's momentum. However this effect is very small and occurs in a period of about 250 million years. Rigid Body Dynamics by A.G. Webster gives the following formula - Angle of precession PiCP 2 = *¥ due to sun's attraction 3ym C - A sin 21 v ip - — £— x cos (o (t - -r— ) 2QT 3 C zn where = y gravitational constant = 6.67 X 10" 8 C.G.S. units C = moment of inertia of earth round the polar axis A = moment of inertia of earth round an equatorial axis o) = Obliquity of ecliptic = 23°26'45" m = mass of sun = 1.99 X 10 33 gms r = distance of earth from sun = 1.49 X 10 cms y m -—- = tide raising term 1 = longitude of the sun n = angular velocity of earth in orbit Corrections to Moon 341 Q = angular rotational speed of earth in radians For a homogenous sphere, C=A and *P = 0. If polar radius C = a (l-£), where e is ellipticity of earth, C - A 1 = e = — if concentric layers of earth C 297 J are assumed homogenous. Bui its real value has been found to be — ~r. Putting the values in formula, 304 ° dWs dt due to sun is ^^ x ^— — cos © (1 - cos 21)= 2.46xl0' 12 rad/sec. Qr 3 C v ' It is multiplied by 2.063 X 10 5 = seconds in radian and 3.156 x 10 7 seconds in a year to get seconds of arc per year. Thus rate of solar precession = 16".0 per year. p m The tide raising force ^^ for moon is more than double of the sun. Thus lunar precession = 34". 4 per year. Moon's orbit is making an angle of 5° 9' average with sun's path (ecliptic) varying ± 10'. Point of interaction of moon's orbit travels on eecliptic in a period of 18.6 years (motion of rahu). Figure 4 shows G, M as poles of ecliptic and of moon's orbit. ,P as celestial pole (earth north pole). Solar precession is vector along line PS. perpen- dicular to CP, lunar precession is represented by vector PR which goes up and down as M goes round C in a cycle of 18.. 6 years (Rahu period) components of motion are 342 Siddhanta Darpana Along PS. = ¥ms = ^s + Wm CosM PC Perp to PS. *Pn = ^m Sin MPC This causes certain irregulaties in precessional motion and also in the annual variation of obliquity - which is called nutation - with a period of 18.5 years If t = no. of years after 1900 AD, then Rate of precession = 50". 2564 + 0" .0002225 t Angle between equator and ecliptic planes is 23°27'8".26 - 0".468 t Correction in precession due to nutation is Figure 4 -17".235 sin (sayana rahu) - 1".27 sin (2 sayana sun) Correction in incline of equator is + 9". 21 cos (sayana rahu) +' 0".55 Cos (2 sayana sun) (2) Indian theories of precession : Correct theories : One theory states continuous backwarct motion which is correct as per modern theory. Other theory indicates -oscillatory motion when is not correct either according to modern theory nor accordiing to references in Vedas or brahmanas. • Corrections to Moon 343 Rates of Steady precession ; Various quota- tions from pur anas, brahmanas indicate different position of equinoxes. Rgveda tells rains from Mrga naksatra (1-161- 13). Taittiriya samhita (17-4-8) indicates vasanta at phalguna full moon. Both indicate a period of 23,720 B.C. when equinox was 352° behind present position. Valmiki Ramayana indicates demon dynasty with Mula naksatra at vernal equinox. This should occur at 17000 B.C. which tallies with Egyptian countings mentioned by Herodotus. It also tells beginning of Dcsvaku dynasty with vernal eqninox at visakha at about 15080 B.C. This was the time of great deluge which is correct as per geology and sumerian records. Mahabharata indicates fall of pole star vega (Abhijit). At about 12,400 B.C. this was the pole star. Hence, around this star, a small extra naksatra had been assumed. Taittiriya Brahmana (1-5-2,6,) states krttika to visakha are Deva naksatras which turn Sun from south. Anuradha to Apabharani are yama naksatra which turn sun from north. This position of winter and summer solstice was in 8357 B.C. Varah- amihira tells that winter solistice was at Dhanistha beginning at time of Vedanga jyotisa and at Makara beginning in his own time (about 100 B.C.). He has concluded backward motion of ay ana. Satapatha Brahmana tells Krttika at equator, present position being 36° 9' east and viksepa 4° 2'. This was about 67° 56' east of present position of vernal equinox. This was 2942 B.C. 344 Siddhanta Darpana J Manjula (932 AD) has indicated backward precession of vernal equinox 1,99,669 cycles in a kalpa i.e. 59". 86 per year. Bhaskara II has also accepted his authority. He has stated that ayana was non existant at time of Aryabhata and negligible at time of Brahmagupta and so they have not discussed. Even Bhaskara has mentioned it only in the context of constructing gola bandha (armillary sphere) Curiously Jagannatha Samrata in his Siddhanta Samrata has indicated 278 Saka as year of zero Ayanamsa and rate of precession per year as 51". This value is accepted as per modern calculations Prthudaka (928 A.D.) has given 56."82 per year. Even Munjala value is very accurate. In 932 A.D. yearly rate of precession was 50.2453- 0.0002225 t (years from 1850 AD) = 50.041". According to Indian practice, excess precession for tropical year is 9.76", then correct precession should be 59.8" per year which is very close to his value of 59".86. Liberation theory : A suspect passage occurs in Surya siddhanta, Triprasnadhikara, (9-10) which states - In a yuga, naksatra cycle falls back eastward thirty scores (f^^TT 30 X 20). Number of days (ahargana) is multiplied by this 600 and divided by number of days in a yuga to give the no. of revolutions and fraction rasis etc. Its bhuja is multiplied by 3 and divided by 10, which will give ayana in amsa or ayanamsa. This gives an oscillatory motion of 27° east and west from equinox point. Corrections to Moon 345 This appears a defective and interpolated passage because- (i) It occurs in Triprasnadhikara and out of context just after discussing directions and shadow lengths. (ii) No where else in this text krtya = 20 units has been used. 30 scores should have been written 6 hundreds or each digit should have been indicated separately through words as per general practice. (iii) The verse indicates oscillation of naksatra cycle around equinox. If it starts with east ward motion; in 5097 years since kali, it should be towards west from equinox. But the 0° of ecliptic is towards east from equinox point, as it has been clearly mentioned in next verse also. Thus the text should have stated oscillations of equinox point around 0° of ecliptic. Due to 600 speed, round number (540) had been calculated at the beginning of Kaliyuga. (iv) Oscillations of equinox within 27° is not mentioned anywhere in ancient texts. They have mentioned the difference of upto 35° and values at different points of time indicates only backward motion. (v) Bhaskara II has quoted Surya siddhanta differently. According to him, surya siddhanta tells 3 lakh backward rotations of Ayanamsa in a kalpa. This means 300 backward rotation in a yuga. This can mean 3 backward + 300 forward = 600 oscillations in a yuga. But this interpratation has not been mentioned in own commentary or any other commentary. Thus he must have mentioned ^ 46 Siddhanta Darpanak some version of surya siddhanta prevalent in his j time. This was lost due to the interpretation! presently found. J ■ > (vi) Reasons of accepting this wrong version is that 0° position is same in both systems around 285 AD. and both indicate backward motion till 2298 AD. Due to approximately -equal angular speed in both system, we get the same position of Ayanamsa. So no body has thought it necessary to refute this theory. Reasons for oscillation theory and its value of constants- (i) Bhaskara and Varahamihira have com- mented that Ayanamsa was zero at the time of Aryabhata 3600 years exactly after Kaliyuga. Now, it has been assumed that all the planetary positions were zero at beginnig of Kaliyuga and they started moving east wards since then. The same assump- tion was made for krantipata which was found west from 0° at the time of Aryabhata. This means that, pata started moving east wards with uniform speed like all madhya grahas, at mid point till time of Aryabhata it started moving backward and reached zero position again. Thus half oscillation was completed within 3600 years. Remaining half oscillation will mean backward motion for 1800 years from Aryabhata and again forward motion for 1800 years, so that it comes to zero position in east ward motion, as in Kali beginning. Thus 1 cycle is 7200 years, giving 600 cycles in a yuga. At about 600 years after Aryabhata, if Ayanamsa was 9° west, then maximum oscillation in 1800 years will be 27° on either side. Such measurement only can be basis of this limit. Corrections to Moon 347 In comparison, Hipparchus (100 BC) had found precession but did not give the value. Ptolemy had estimated it to be 36" per year. Albatani of Arab in about 880 AD, found the speed as 55". 5 Then Nasiruddin of Iran calculated in 1250 AD as 51" per year which was very accurate. Siddanta Darpana has assumed surya siddhanta theory of oscillation, but has slightly corrected the value to 6,40,170 oscillations in a kalpa instead of 6 lakhs for a kalpa according to Surya siddhanta. These corrections are based on the following- (i) Assumption of true 0° position which is with 1/2° error in eye estimates- This is according to position of identifying stars as given in Surya siddhanta. This will indicate, the current value Of ayananasa as to how much vernal equinox has shifted west from this 0°. (ii) Assumption about the time of 0° ayanamsa- it is clear that surya siddhanta value is based on ° ayanamsa at the time of Aryabhata in 3600 kali in which half oscillation was complete. Figure of 6,40,170 oscillations in a year by siddhanta darpana indicates 0° ' ayanamsa in 284 AD. At present it is assumed to be on mesa sankranti of 285 AD. So reasons of Candrasekhara must have been same as current reasons for accepting this figure. It may be noted that both theories give same figure at present because, their speeds are almost same. 27°X4 = 108° oscillation in 7200 years means 1° in 66.6 years. Siddhanta Darpana gives 1° in 61-4 years. Modern figure is 72.24 years per degree f or AD and 71.63 years at 1900 AD. Munjula 348 ' «*■ ' Siddhanta Darpana figure also is 1° in 61 years. This was accepted by Bhaskara and this figure only has been accepted by Candrasekhar though under different theory. (3) Formulas explained : Revolutions of Ayana till desired day _ Ahargana Revolution in a kalpa " No. of days in a kalpa In a full revolution of 360°, quadrants are of 90° each. In oscillatory motion the corresponding quadrants are 8 - 90° 0° to + 27° 90° --180° + 27° to 0° reverse motion 180 ' - 270° 0° to-27° reverse motion 270° to 360° -27° to 8 forward motion Thus 27° Ayanamsa = 90° revolution 27 or Revolution X — is Ayanamsa. 27 3 60 Hence revolution is multiplied by —= — or -r-r which has been mentioned here. Verses 92-99 - Calculation of Kranti Planetary orbits (ecliptic) and equator circle, both are in east west direction. Due to inclination, they cut each other which results in kranti (north south deviation). Thus, deviation of the planet, north or south from equator is measured along great circles passing through north pole and south pole (of earth projected in sky). This is also called 'apama' or 'apakrama'. Note : Kranti (apama or apakrama) is north south deviation from equator ar seen from earth. '1 ■ J s Corrections to Moon ^49 Sara or viksepa is north south deviation from ecliptic as seen from sun. Both are measured along great circle perpen- dicular to reference circle (equator or ecliptic). In celestial sphere, an imaginary circle of rotation of sun is called kranti vrtta or marga (ecliptic circle of path.) It is divided into 12 rasis. Ayana correction is done in 1st and 7th rasis (0° and 180° position). The corrected positions of these rasis give the positions of intersection of ecliptic with equator circle. These points are called pata. Since day and night are equal, they are called sampata. Thus there will be two sampata, vasanta and hemanta (vernal or autumnal equinox).. At 3 rasis from sampata, kranti will be maximum (23° 30') in north or south directions. Jya of maximum kranti (23° 30') is (1370'). Graha position corrected by ayanamsa only is used for calculation of bhujaphala and jya. Ayanamsa is added to spasta graha, sum (sayana graha) is multiplied by Jya of parama kranti (1370) and divided by radius (3438). This is equivalent to multiplication by 100 and division by 251. This will be kranti jya of the spasta graha. This value converted to arc will give kranti in kala. Square of kranti jya substracted from square of radius (1,18,19,844) and taking square root gives 'dyujya' which is half diameter of ahoratra vrtta (diurnal circle) - explained in Triprasnadhikara. Kotijya of sphuta graha (corrected with ayanam§a) multiplied by 100 and divided by 251 and multiplied by daily motion of graha gives daily motion of kranti. 350 Siddhdnta Darpatui Ayana corrected graha moves northwards in 1st and last quadrants and south wards in 2nd and 3rd quadrants. Notes : (1) Kranti from sayana graha - '.:B A 0° P : O is the O" of ecliptic. By definition, kranti at point A of intersection of equator will be zero, because it is at equator also. A is towards west from due to backward motion. Planet P on ecliptic is counted in east direction from 0° of ecliptic. Thus kranti of planet P increases from A in the east direction, where it is zero. Thus sayana graha AP = OA (ayanarhsa) + OP (distance from 0° of ecliptic i.e. true graha). (2) As seen from equator, the pata, A where kanti vrtta appears moving north wards is the pata taken as 0°. Ectiptic ■Wl '1 ■■■.1 B ■ h Equator J! Figure 5 | In figure 5, AE BA' is equator and AB' A' is j ecliptic which cut each other in line A O A'. OBJ and OB' are radius perpendicular to AA' at O -■VJf: Corrections to Moon 351 which is point of observation at centre of celestial sphere. This equator and ecliptic are inclined at an angle B'OB which is about 23-1/2° Position of planet is at P on ecliptic whose distance from point A is the sayana graha = AP. PE is arc of great circle perpendicular to equator, hence passing through pole of earth or equator. Thus length PE is the kranti, which can be determined from relations of right angled triangle APE on the sphere. Hence sayana graha AP needs to be calculated to complete this triangle. According to Napier's law for right angled spehrical triangles, sine of middle part = product of cosines of opposite parts For middle part taken as PE, opposite parts are | - PA and | - ZPAE Hence sin (PE) = sin (PA) X Sin (PAE) ~ v *. . - _ . _• RsinPA'x R sin PAE or Kranti lya = R sin PE = iJ R Here PA = sayana graha, Z.PAE = parama kranti Hence, krantijya = Jya of sayana graha x Jya of parama kranti Trijya Thus the position of highest kranti B' is at from A of 0° kranti. Another point of highest kanti is opposite to B' i.e. 90° from A'. s - Jya of parama kranti _ 1370 100 Trijya = 3438 = 251 a fixed quantity, hence alternate formula has °een given 90 352 Siddhanta Darpan (3) Speed of kranti : Sin PE = Sin PA x 100 251 . .'j* Differentiating both, cos PE. d (PE) = cos PA,] 100 j d(PA).— For a single day, point A can be consider© fixed and d (PA) = d (AO + OP) = d (OP) = spee of nirayana graha as d (AO) = for small period PE is small and cos PE can be taken almos equal to 1. So speed of kranti is d(PE) 100 = — Xd (PA) cos PA = 152 X speed of eraha X cos of sayana graha 251 ^ . • Verses 100-101 : According to Bhaskarcarya, ayana doesnl move in west direction, hence he has asked to ad^ ayanamsa to the graha always. Still according t Brahma and siirya siddhanta, I have assumed i\ motion in both directions. It will be clear by calculating ravi from chaya (shadow of gnomon). Verses 102-104 : Day night values at a place-Kranti jyi multiplied by palabha (shadow length of 12 len. stick on equinox day) and divided by 12 giv ksitijya. This, multiplied by trijya (3438) *i divided by dyujya gives carajya. Its arc will be prana. Caraprana added to \ of day night (15 danda| gives half day length when it is north kranti. Or an< car* Corrections to Moon 353 abstraction from 15 dandas, half night length is obtained. When kranti is south, opposite procedure is followed - day half is obtained by 15 - caraprana and night half = 15 + caraprana. Multiplying them by 2 we get values of day and night (Quoted from surya siddhanta) For finding day and night periods of naksatras, moon and other planets, their sara is added to kranti, when they are in same direction, or difference is taken, when they are in opposite direction. From this spasta kranti, day or night time is found, (Day time is the period for which planet is above local horizon) Notes (1) These topics have been discussed in Triprasnadhikara, but to understand the mean- ing of these formulas, it is necessary to explain the technical terms. On equinox day, sun is perpendicular on equator, hence at local noon on an equator place it will be directly above, i.e. perpendicular to horizontal plane. Hence, a perpendicular to horizontal plane at other place with latitude 0, will be at an angle <p with sun's highest position at noon. Thus the length of a vertical pilllar's shadow at noon time on equinox day will give latitude of the place. N C A E' 354 Siddhanta Darpan In Figure 6, S is perpendicular on equator; passing through E, S being direction of sun. Aif place P, latitude = 0. Hence direction of sun i CA direction makes ZCAP = <p with vertical direction of pole PA = 12 unit length. m Tan 6 = PC PC AC " 12 ...*J gives measure of latitude <p Figure 7 - Calculation of day time at a place Figure 7, is a diagram for place O where day length of a planet i.e. period for which it is above horizon is to be found. NOS is horizontal line in north south direction at that place and DOD' is the horizontal line for equator. D is celestial north pole (direction of earth's north pole in the sky) and D' is south pole of earth. N D V S is the north south circle and V is the vertically upward point at O. Due to daily rotation of earth, planets appear to move in circles parallel to equator. These circles are called ahoratra vrtta (diurnal circle). For different positions of a planet or naksatra, the Corrections to Moon 355 circles projected on vertical plane are P1P1', P2P2 7 and P3P3' all parallel to equator P2P2'. Sun on equinox day will appear moving on P2P2' - kranti for short time assumed constant. When north kranti of a planet is arc P2 Pi' then its diurnal circle is P1P1'. When south kranti is P2 P3, then the circle is P3 P3 7 . (diameter only is shown in projection). At equator, the horizontal line DOD' cuts all the diurnal diameters in two equal parts. As long as the planet is above horizon or on V side of DOD', it is seen or rising. Below it; it is set. Thus at equator, day and night are always equal. However, for place O, the horizontal line is SON. Day portion of the planet is PiH or P3H'. It is bigger than 12 hours for north kranti. OV is radius, PiK =Dyujya (diameter of diurnal circle) P1P2 arc or L PiO P 2 is kranti Hence, kranti jya = PiL = OK Versin of kranti = P2L (versin = 1- Cos 0) Dyujya = PiK = OL = OP 2 - P 2 L = Trijya - versine of kranti = Kranti koti Jya (1) Ksitijya = KH (extra motion on diurnal circle beyond half day). Latiude <p = Z.HOK or L VOP 2 HK In AKOH, tan (j> = — OK _ Palabha But tan <p - —7Z — T 12 356 Siddhanta Darpang^ „,. . . ,,„ Kranti jya x Palabha Hence Ksiti-jya KH ^^ - - (2) | But DKO and DHC both are perpendicular! on P1P1' and P2P2' (in the spherical triangle). I Pi K P 2 O 1 Hence = 4 KH OC nr nr- *** X F2 ° Ksi%a x Trijya ,§ or oc = ~ Sk = 3^T " " ■ (3 >l This is value of OC = carajya. Its angular! kala value is caraprana, because earth takes 1 prana | to rotate kala. From the equations (1), (2), (3), Carajya Kranti jya x Palabha Trijya 12 Kranti Kotijya Krantiiya Palabha _ .. x „ x Tn jya Krantikotijya 12 = Kranti sparsa jya X Aksansa sparsa X Trijya (4) In modern terms when Kranti is 6 Sin (cara) = tan <p tan 6 (5) Complete day is rising from horizon H to top position and then coming back to M again, after 1 which it sets. Hence half day = — day night + carajya. Verse 105 : Correction due to sara in day time From surya siddhanta when kranti and sara are in one direction they are added to find spasta kranti of a planet (true declination from equator). When they are in opposite direction, their dif- ference is taken for spasta kranti. Notes : Kranti is inclination of planet from ecliptic. It is caused by two angles - Angle of Corrections to Moon 357 ecliptic with equator whch is called kranti (mean value). However, a planet deviates from ecliptic, whose angle is known as sara. Hence total inclination with equator is sum of these angles. This inclination only, decides their day and nights. Verses 106-112 : Easy calculation of cara - Now a rough practical method is described to find out cara in pala. (i) Find out the cara kalas at the end of 1,2 and 3 rasis (corresponding to their krantis) (2) 3rd cara khanda = 3rd rasi cara -cara of 2nd rasi 2nd cara khanda = cara of 2nd rasi - cara of 1st rasi 1st cara khanda = cara of 1st rasi itself ■ • These are the cara of mesa, vrsa and mithuna rasis in reverse order. (3) Bhuja of sayana planets is taken, its rasi and degrees etc. are kept separately. If it is less than 1 rasi, then degree and minute (kala) are kept separately. (4) Degree and kala are multiplied separately by cara. Result at kala place is divided by 60, quotient in degree added to degree place, remainder to be kept as kala. Total degrees are divided by 30, remainder is kept there and quotient is added with rasi. If bhuja is more than 1 rasi; but less than 2 rasi, position is multiplied by 1st cara for mesa rasi, kala and degrees are multiplied by cara khanda of 2nd rasi. As before, excess kala and degrees are added in higher places of degrees and rasi. 358 Siddhdnta Darpanti If bhuja is more than 2 rasi (it will be always less than 3 rasi). then 1st and 2nd carakhandas are added at 1st place of rasi. Degrees and minutes are multiplied by 3rd cara khanda. These are converted to rasi, degree, kala as before. Alterna- tively, cara of each rasi of ravi is taken and accordingly, their fraction for each degree is calculated. Notes : Rationale of method is obvious. It is linear interpolation which assumes that variation rate of cara within a rasi (30° interval) is constant. This gives some error which can be ignored for practical purposes. Verses 113-117 : Udayantara pala from sayana surya - Now method to find udayantara sanskara is being explained. This is difference in pala between true sunrise time and madhyama sunrise time at Lanka. This is called time equation, arising out of inclination of ecliptic with equator. This rises steadily in first 3 half rasis (i.e. 3 X 1/2 X 30° = 45°) and decreases till next 3 half rasis. From the first sampata point, udayantara (in pala) rises by, 12, 9, 4 pala for first 3 half rasis, From 4th half rasi to end of quadrant it declines by same amounts 4, 9, 12. We find out the udayantara palas for com- pleted half rasis. Fractional portion of lapsed degrees is multiplied by pala of that 15° part and is added to the result for completed half rasis (if udayantara pala is rising). It is substracted if udayantara is declining. i : -a Corrections to Moon - 35 ^ When bhujansa of surya is 45° (3 X 15° or 3rd half rasi), its udayantara pala is maximum 25 palas (12 + 9 + 4 pala). After that it starts declining. On equinox day or at 4th rasis from that (0°,90°, 180° or 270°) udayantara pala is zero. Udayantara pala is multiplied by daily motion of graha and divided by no. of pala in a day (3600). Result is added to madhyana graha, if bhuja of sun is in even quadrant, otherewise it is sub- stracted. Result will be the graha for sunrise time of Lanka. Notes : This is approximate udayantara palas at the end of each half rasis. Its complete explaination will be given in Triprasnadhikara. (p/447) Verses 118-120 : Rising time of planets - We add ayanamsa to graha, and from sayana graha its udaya time in asu (prana = 4 seconds) is found. Udaya asu is multiplied by daily motion of the planet and divided by no. of kalas in a rasi (1800) Result is added to kalas in a circle (21,600), if the graha is margi (moving forward). If vakri (retrograde), it is substracted from 21,600. The result will be day of the graha in asus i.e. after 1 rise, it will rise after that time again. Savana dina for sun is roughly 60 danda. 59 lipta less from that is a naksatra dina. Method for finding savana dina of a planet has been told. Notes : Udaya asu of a graha is its rising time, as its speed is seen from an inclined plane which will be less than its speed in the ecliptic. This will be less than its normal rising time. The corresponding apparant speed is found by dividing ■■.a d Ml 360 Siddhanta Darpa% the rising lime of that rasi by 1800 kala and multiplying it by gati of graha, this is movement in one day as seen from a latitude. If graha i$ moving ahead, this will be extra time taken earth to reach its next rising place. Hence this timelj is added to 21,600 asu. | Verses 121-126 : Rising time of rasis The rasi which rises (on eastern horizon) at a time is called lagna. At sunrise time, rasi, arhsa etc of surya itself is lagna. Savana day night (ahoratra) is found from daily motion of ravi as explained above. From true ravi at desired time, current rising rasi (lagna is found). In l/12th part of kranti vrtta (rasi), there are 1800 kala. Near equator, their inclination to equator is more. At the end of ayana (south or north), i.e. 90° east or west from equinox, kranti vrtta (ecliptic) is paralled to equator. Hence, in diurnal circle (parallel to equator), different parts of ecliptic rise in unequal times. To find out the rising times of rasis at equator, the jyas of 1,2 and 3 rasis and their kranti jyas are squared separately. For each rasi, kranti jya square is deducted from jya square, and square root of the difference is taken. These are multiplied by trijya and divided separately by jya of 1,2,3 rasis. Arc of the three results is calculated. Rising time in asu for third rasi is found by substracting arc of 2nd from 3rd rasi. For rising of 2nd rasi, arc of 1st is substracted from 2nd rasi. Arc of 1st rasi is its own rising time. Thus we get rising times in asu (udayasu) of (1) mesa (2) vrsa and (3) mithuna rasis. Udayasu Corrections to Moon 361 of next 3 rasis are in reverse order i.e. rising time of (4) karka is same as of 3rd rasi, of 5th sirhha it is same as of 2nd rasi and of 6th kanya and 1st is same. Rising times of tula to mina is in reverse order of the times for 1st to 6th rasis. Comments (1) Steps in calculation Jyas of the rasis (1,2,3) or 30°, 60°, 90° are 1719, 2978 and 3438 Their squares are (29,54,961), (88,68,484) and (118, 19,844) Kranti jya of 3 rasis are (685), (1186), (1370) Their squares are (4, 69, 225), (14, 06, 596) and (18, 76, 900) Substracting kranti jya squares from jya squares, we get (24, 85, 736), (74, 61, 888) and (99, 42, 944). Square roots of these results are (1576/37), (2731/39) and (3153/15) They are multiplied by trijya (3438). Products are (54,20,408), (93, 91, 413), (108, 40, 873) They are divided by respective dyujyas (3369), (3227), (3153) Results are (1609), (2190) and (3438) Their arcs are (1675), (3471), (5400) Rising time of mesa = 1675 Rising time of vrsa = 3471-1675 = 1796 Rising time of mithuna = 5400-3471 = 1929 These are better approximations for modern values for 23° 27' inclination of equator. These are 362 Siddhanta Darpana based on 23 ° 30' declination and old siddhantas assumed 24°. Comparison is given below Modern values Sayana rasi Rising time in asu in old siddhanta Rising time in Siddhanta Darpana Mesa Vrsa * • Mithuna 1670 1795 1935 1675 1796 1929 Asus Minutes 1675 111.7 1794 119.6 1931 128.7 Lanka rising time for all rasis (Siddhanta darpana) Value 1675 1796 1929 rasis rasis rasis rasis (1) Mesa (6) Kanya (7) Tula (12) MIna (2) Vrsa (5) Simha (8) Vrstika (11) Kumbha (3) Mithuna (4) Karka (9) Dhanu (10) Makara (2) Derivation of rising time formula for 3 rasis Figure 8 - Rising times of rasis at equator Figure 8 is horizon circle of equator in which E, N, W and S are the points in east, north, west and south. WOE is equator circle K'OK is ecliptic projection . O = Vasanta sampata (or vernal equinox) Corrections to Moon 363 N is also direction of north pole of earth. Daily rotation of earth is along circle WOE, the time in which OE part of equator rises, is the time of rise of OK part of ecliptic also. But rising time of the whole equator circle 360° is 1 naksatra dina (sidereal day) which is equal to 21,600 asus by definition. Hence rise of 1 kala on equator will take 1 asu. Hence length of OE in kala will give the rising time in asu which is rising time of OK part of ecliptic also. OEK is a spherical triangle in which ZOEK is right angle, ZEOK is angle between equator and ecliptic which is maximum value of sun's kranti. EK is kranti of point K, arc OK is sayana rasi of point K measured from equinox point O. OE is its length measured on equator (visuvansa). Hence as per Napier's rule - Cos KOE = tan OE x cot OK cos(parama kranti) or tan OE = ——. -,. N - - \\) cot (sayana rasi) For finding values in R sines (jyas), relations in spherical triangle NOK, SinNK _ SinOK sin NOK " sinONK But L ONK = arc OE SinNK sin OK Hence sin NOK SinOE sin OK X sin NOK or Sin OE = . XT1 , sin NK Here OK = sayana value of K 364 Siddhdnta Darpana L NOK = ZNOE - L KOE = 90°- Parama kranti of sun Hence Sin NOK = cos (parama kranti) = Dyujya of 3 rasis (because cos (kranti) = Dyujya) Sin NK = Sin (NE - KE) = Sin (90 ' - kranti of K) = cos (kranti of K) = Dyujya of K Thus Sin OE Sin (sayanaK) x Cos (Parama kranti) cos (kranti of K) * ' Alternatively it is, Sin OE Sin (sayanaK) x Dyujya of Parama kranti Dyujya of K (3) Formula (3) has been given in the next verse. In . spherical triangle KOE sin KE sin OK ~ — ix^TT = ~ — ^T^7 = sin OK ( as sin OEK = sin KOE sin OEK v sin 90° =1) Thus in formula (2) Sin OE Vl - Sin z KOE = Sin OK x D yu jya of K sin OK , _ v 2 1 sin Ke x sin OK K Dyujya Vsin 2 OK - sin 2 KE K dyujya or R sin OE= V (R sin Ok)' - (R sin KE)* Dyujya of K y) Corrections to Moon 365 This is the formula described in this verse. (3) To prove that rising times of 4th to 6th rasis are equal to those of 3rd to 1st rasis in reverse order - Equation (3) above tells sin OK x cos (Parama kranti) Sin OE = ^=t cos (KE) OE = rising time or length on equator in kala. OK = sayana rasi of K, KE = Kranti of K. Sin = Sin (180°- 0) Hence Sin (180° - OE) Sin (180° - OK) x Cos (parama kranti) cos KE Rising times of 90° at equator or ecliptic are same i.e. when OK = 90% OE == 90°. For rising time of mithuna (60 o -90°), we substract the rising time of 60° from 90° time (6 hours = 15 danda = 5400 asu). Rising time of 180° also is equal on both circles as it is equal for every 90 ° . Hence, rising time for karka (90° to 120°) is found by substracting the time of rising time of 3 rasis from 120° time. Now, when OK = 60°, OE is rising time (slightly less than OE) When OK = 120° = 180° -60°, its rising time = 180° - OE Hence rising time *>f Karka = (180°-OE) - 90° = 90' - OE = rising time of mithuna. Similarly we can prove that rising times of simha, vrsa and kanya, mesa are equal. 366 Siddhdnta Darpana The rising times of rasis from mesa to kanya are equal to tula to mlna in reverse order for all palces, not only on equator. So this result will be proved when rising time at other places is calculated. This is evident because both the ecliptic and equator circles bisect each other, hence other half 180° to 360° is similar to 180° to 0°. Verses 127-128 : Alternative method for rising times at equator Dyujya of 3 rasis (3153) is multiplied separate- ly by jya of 3,2, 1 rasis (3438/2978/1719). Results (10,840,014), (93,89,634), (54,20,007) are divided by dyujya of the rasis (3153, 3227, 3369). Arc of the resulting ratios treated as jya is found (5400, 3471 and 1675), which are rising times of 3, 2 and 1 rasis. Rising time of 2 rasi is substracted from 3 to give time of 3rd rasi. Time of 2nd rasi is time of 1st rasi deducted from rising of 2 rasis. Rising time of 1st rasis is already known. Notes : This method has already been proved in previous verse. Verses 129-130 : Rising times at different parts of sky. Rising times of six rasis in asu or prana have already been stated as (1) 1675 (2) 1796 (3) 1929 (4) 1929 (5) 1796 and (6) 1675. (These have been calculated for rising on east horizon on equator). The rising time of rasis for other points (on the east west vertical circle) are also the same. These points are udaya (east horizon), Asta (seetting point in west horizon), Dasama (Tenth house or vertically Corrections to Moon 367 upward point), Caturtha (fourth or vertically down wards). Note : This is because all quadrants are same on both circles. Veerses 131-142 : Lagna at any place - To find .out rising times of rasis at other places, we find out the cara khanda of first three rasis as per formula described for that place. These cara khandas are deducted from first and last 3 rasis in that order and are added to the three rasi from karka and in reverse order to three rasis from tula. Addition and deductions of carakhandas is to the rising times of rasis at Lanka (both in asu or prana). These give the rising time at other place, for which carkhanda had been calculated. • ■ According to rough calculation, whatever rasi is rising in east horizon, its seventh rasi (180° away) is setting in the west. Rising times for each hora (1/2 rasi = 15°), or dreskana (1/3 rasi = 10°) can also be calculated in same manner. For that kranti and dyujya is calculated for each half or l/3rd rasi, hence it will be more accurate, than rising time for rasis. Ayanamsa is added to sun at sunrise time position. Lapsed and remaining parts in the incomplete rasi of sayana sun is calculated. Remain- ing degrees of the rasi are multiplied by rising time of full rasi and divided by 30. This gives rising time of remaining part of that incomplete rasi. This is substracted from desired time interval after sunrise (called ista kala). From the remainder, rising times of next rasis in successive order are deducted. Last remainder from which rising time ''■'■■$ Siddhdnta DarpcajM 368 of next rasi cannot be deducted - is multiplied by 30 and divided by rising time of the next rasi. This ? result in degrees etc. is added to the completed > rasi which has risen. This gives sayana lagna. Ayanamsa is deducted from this to give the lagna for required time at desired place. When fractional rising time of remaining rasi of sayana sun is more than ista kala, the same sayana lagna will continue to rise at ista kala. This remaining rising time is multiplied by 30 and divided by rising time of sayana sphuta ravi (or roughly by rising time of that rasi). Result in degrees etc is added to sayana sphuta sun and ayanamsa is deducted to find sphuta lagna. To find the moment when a particular lagna will rise, ayanamsa is added to it. Its lapsed part in incomplete rasi is multiplied by rising time of that rasi and divided by 30. This is lapsed rising time of the fractional rasi. To this, we add the rising time of remaining fraction rasi of sayana sun at sun rise time, and the rising times of next completed rasis upto the completed rasi of sayana lagna. The grand total will be ista kala after sunrise, when the desired lagna will rise. Wk £ Figure 9 - Rising times of risis at places other than equator Corrections to Moon 3&g Note : (1) Rising times for a place of latitude <p NESW is the horizontal circle at desired place of latitude q north, (fig. 9) P is the north pole in sky : O is vernal equinox point. WOE is equator circle, KOK' is ecliptic circle. EC = Cara of K which is below horizon. When point O is rising on east horizon, sayana 0° of both ecliptic and equator are rising. When K point of ecliptic rises on horizon, E point on equator also rises. Hence, rising time of OK in asu is same as that of OE. The polar circle PK passing through K meets OE extended at C which is below horizon. Thus OEC arc is the rising time at equator for point K. Hence rising time at 0° North is found by deducting EC from rising at equator. EC is the cara-kala for point K. Thus rising time = Equator rising time - carakala. Cara jya = R tan <p x tan where is kranti of K. It has already been proved after verse 103 For mesa rasi, OK = 30° Kranti of K is KC Carajya = R tan KC X tan <p = EC Rising time OE = OC - EC This holds good for mesa to mithuna i.e. 0° to 90°. For karka rasi, OK = *120\ Then kranti of K is same as of vrsa rasi i.e. KC is same. Hence EC is also same as for 60° (vrsa). Hence rising time of karka = OE = OC - CE '"■ * k 370 Siddhdnta Darpa$§ =(Rising time, of 3 ra&s + karka) - cara otj Vrsa = (Rising of 3 rasis-cara of 3 rasi) + karka + (cara of 3 rasi - cara of vrsa) = (rising time of 3 rasi at lat) + karka + j cara of mithuna Hence extra rising time for karka = karfc*| rising at equator + mithuna cara. | Similarly cara of vrsa is added to sirhha, ami | mesa cara is added to kanya rising time at equator.^ E 1 i 1 A rH Figure 10 - Rising limes for tali to mira (2) Rising times for rasis tula to mina - Tg fieure 10 for 2nd half of ecliptic is same but the difference is that the two circles after crossing ead* other at autumn eqiunox O, have .« versed a *^ positions. K'O part of ecliptic which was abovj equator till 180' at O sayana, goes below equate* after O at OK i.e. after sayana tula. Hence, tui* to 3rd rail from it, cara portion CE is to be a to the rising times at Lanka. Thus ^."""SJT = tula time at equator + cara of 1st 30 (mesa}. i 1 Corrections to Moon 371 This is same as rising time of kanya as proved in previous section. Similarly times of vrscika and simha are same and so on in that order. (3) Calculation of lagna - OE is east horizon at sunrise time and OE' is its position at ista kala after sunrise. Ai, A2, A3 A7 are successive positions of starting of rasis. (fig. 11) At sun rise time, point E is on east horizon and lagna, and sun also is rising at E. Hence, at sunrise, rasi of sun and lagna is same. The rising time of E' is the sum of rising times of EA2 (remaining part of fractional rasi A1A2) then rising of complete rasis A2 A3, A3 A4, A5 A6 and then lapsed part of fraction rasi Ae E'. A . 1 E A2 A3 Figure 11 Calculation of Lagna Within a rasi the rising times can be con- sidered as proportional to the parts, hence Rising time of E A2 _ degrees of EA 2 Rising time of Ai A2 30 ° of Ai A2 Similarly rising time of completed part A6 E'can be calculated as fraction of rising time of rasi AeA 7 . 372 Siddhdnta Darpm Since the rising times are not proportional t rasi length (mesa rising is much faster than vrs for example) this calculation will be more accurate, if rising times of smaller parts like hora = 1/2 or dreskana = 1/3 rasi are calculated. • ■ Verses 143-151 : Rule for finding das lagna : Madhya lagna or tenth lagna (vertical to position) is found by rising time of rasis at equat only for all places. (Because south north line bisec the diurnal circles at all places and corresponding times are same at any place and equator). Before mid-day, the period for which sun remain in east is called nata kala which is desire time before mid day. Nata kala in east direction is expressed in asui From this, we deduct the equator rising time o J completed part of sayana sun rasi. From remainder! the equator rising time of previous rasis i' :J substracted successively. Last remainder is divide by equator rising itme of next rasi (whch cann be deducted from remainder) and multiplied b; 30. Result in degrees etc is substracted from 30" This is added to the previous rasi which become sayana madhya lagna. Spasta madhya lagna found by deducting ayanamsa. When nata kala is west, the time passed aft mid day is nata kala. From this we deduct the fractional rising time of sayana sun at equator. Then rising times of next rasis are deducted. Last remainder is divided by rising time of incomplete rasi and multiplied by 30. Result in degree etc. is added to completed rasis to give sayana madhya lagna. From this, ayanamsa is to be deducted. Corrections to Moon 373 When in purva or pascima nata, nata kala is less than the rising time of fractional rasi (lapsed or remaining, then nata kala is divided by rising time of the rasi and multiplied by 30. Result in degrees etc is substracted from sayana ravi for purva nata and added to it for pascima nata. Sphuta sayana surya at midday is the madhya lagna at that time. pi w P3 L^P4 P2 V N X P5 B r ■ 'Aa B' *A" P6 Figure 12 Notes : (1) Figure 12 is the vertical circle of any place O. E and W are east and west points. T is the top most or vertically upwards point. D is opposite to T and down ward point. Earth is rotating in clockwise direction, hence ecliptic appears moving in anti clockwise direction - shown by arrow. Movement of ecliptic is not in this plane and °nly its projection is considered. At sunrise time, its projection will be at E which is not at 90° from D, it is upwards for north latitudes when kranti °f sun is south. ETW is the day position and WDE is night portion of sun. DET is position of purva n ata (mid night to mid day) and TWD is pascima nats. Vertical point T is same for all places, because 374 Siddhanta Darpan apparent rotation wil be paralel to equator. Heri rising times at equator are taken. For a position of sun at A in purva nata, thj tenth lagna is lagna at point T. Pi,P2 - - - P6 ajp the start of successive rasis which will rise on after other in clockwise direction. Thus the rasi sun at A will reach T after travelling AT portioj| Current rasi at T is less than A. For point B or WM at pascima nata, rasi at B has aleady risen at T % hence current rasi is more than sun's rasi at ; B. | Position of sun at A or B indicate the time. Thus, for purva nata of sun at A, tenth lagna at T is = A - AP 5 - P5P4 - P4T For pascima nata, Sun is at B, tenth lagna at ; T is = BP 2 + P2P3 + P 3 T. For calculation of rising time of part rasis, the rising time is considered proportional to degrees within the rasi which is roughly correct. Verses 152-153 - Rising time of Nirayana rasis - when ayanamsa is moving eastward, we take the difference of rising times of desired rasi (sayana value) and next rasi, it is multiplied by ayanamsa and divided by 30. If rising time of next rasi is more, then the result is added to rising time of sayana rasi to get the rising time of nirayana rasi. If next rising time is smaller, it is subs tr acted. When ayanamsa is moving west wards (which is not the current position), we take the difference of rising times of the desired rasi (sayana value) and the rising time of previous rasi, it is multiplied by ayanamsa and divided by 30. If the rising time of previous rasi is more, result is added to sayana Corrections to Moon 375 rising time to get the rising time of nirayana rasi. If previous rasi time is more, it is added. Notes : Method is obvious. Since ayana is moving east wards, sayana rasi is more. Hence, rising time of nirayana rasis will be found by comparison with the next rasi. Assuming ayanamsa of 23°, nirayana mesa 0° = sayana mesa 23°, nirayana mesa 30° = sayana mesa 53°. Hencewe have to find the rising times at 23° and 53° at sayana value, and their difference is rising time for mesa. Verse 153 - When ayana (ecliptic) is moving west wards (from point of equinox) then rising times of cara rasis (1, 4, 7, 10th rasis) is same for sayana and nirayana values at equator. At other places also rising times of mesa and tula will be same for sayana or nirayana values (Their previous rasis have same values). When ayana (ecliptic) is moving east from equinox point, then dvisvabhava rasis (3, 6, 9, 12th) rasis have same sayana and nirayana rising times at equator. At other places, only the 6th and 12th rasis have same rising times for sayana and nirayana (as the rising times for next rasis are same). Verses 154-156 : Rising times for Orissa (22 °N) and equator Rising times for sayana rasi at middle of Utkala (22 °N) in danda pala etc are as follows - mesa (3/56), vrsa (4/24), mithuna (5/7), karka (5/37), simha (5/34), kanya (5/22). For second half circle starting from tula, values are in reverse order. w ~ • • «376 Siddhanta Darpatut Udaya times for 22° Ayanamsa - for nirayana rasis are mesa (4/17), vrsa (4/56), mithuna (5/29), karka (5/35), sirhha (5/25) kanya (5/22), tula (5/31), vrscika (5/36), dhanu (5/15), makara (4/35), kumbha (4/3), mina (3/56). Nirayana rising times at equator are mesa (4/54), vrsa (5/16), mithuna (5/22) karka (5/35), simha (4/44), kanya 4/39), Values from tula etc will be in reverse order. There values will change with change in ayanamsa. Veerses 157-160 - Values and Charts Values of 28 naksatras have been stated here according to sages like Garga and Vasistha. Extent of rasis (30°) is clear. Dhruva (constants) have been stated in lipta approximately. Motion of pata of equator and ecliptic also have been given in appendix for 73 values of days. From them, ayanamsa for any day can be calcualted. Motion of pata of equator and ecliptic in a day is liptas etc 0/0/31/32/51/35/6/53/28/23. On first day of kali, visuva sampata was 702 liptas from fixed mesa 0° (towards east). In Karanabda beginning, kranti pata in rasi etc. (for 1869 AD) was 3/16/33/47/27/40. To find out the ayanamsa since kali beginning easily, we find the years since kali beginning according to madhyama siirya. (219/19) is deducted from it. Result is multiplied by 100 less (i.e. 2416) and divided by 2516. We get result in kalas. By '4 Corrections to Moon 377 dividing with 60 we get degrees. Hara of 54° is substracteed from it. Notes : (1) Kranti pata and ayanamsa are considered same thing. But in terms used in the book, kranti pata always moves in reverse direction, making complete circles of 360°. But ayanamsa is 3/10 of its bhuja calculated according to quadrant of the kranti. Kranti at kali beginning can be calculated by multiplying kalpa bhaganas by 1811 and dividing by 4000. (chapter 3, verse 51). Thus bhagana at kali beginning 1811 = 640170 x -— — = 2,898,36. 9675 revolutions 4000 This is 0.0325 revolutions less than complete revolutions. Since pata in moving in backwards direction, it is 0.0325 east of 0°. Thus Kali position is 0.0325 revolutions = 0.0325 X 360° X 60 lipta = 702 liptas. One revolution is in 432 crore years of kalpa divided by 640170 revolutions i.e. 6748.207507 years. Hence pata will come to 0° in reverse motion in 6748,207507 m 360 x 60 19 ., = 219 — Years approx. 60 Hence revolutions at kali beginning are counted from 219/19 yeears after kali (position of 0°). In Karanabda 4971 years had been completed since Kali. In about 3374 years half revolution will be complete in 3374+219 = 3593 kali year. Remain- 378 Siddhdnta Darpana ing years are 1378 years in which less than 1 quadrant will be covered. Thus with reverse motion pata crossed 4th and 3rd quadrants in half cycle and is now in 2nd quadrant at the end of which (i.e. at end of 1st quadrant in forward motion), kranti will be 90° correspondign to - 90° pata or + 27° ayanamsa. By this method the Karanabda pata is about 3 Y 16° 34' approx. Ayanamsa is less than 3rd rasi position of 27° by 16° 34' x 3/10 i.e. 27 '-4° 58.2' = 22 °1' approx, which is given at the end of verse 83. (2) Ayanamsa = (y-210/19) X 2416/2516 Kala X 1/60 degrees. 219/19 years are deducted because, in that kali year kranti pata and ayanamsa were zero. As the ayanamsa had become zero after 1/2 revolution of kranti 27°X2 = 54° movement of ayanamsa in 2 quadrants, this amount is substracted, called hara. . . ,. 108 x 60 Movement per year in kala = ,„ Mn „ MM „ ^ J 6748.207507 = 0.960255 Kala 0.960255 X 2500 = 2416.0016 2416 Hence annual movement of ayanamsa = Kalas Thus we get the formula. Verse 160 - Charts have been given in appendix for Jya (R sine) for 24 khandas of 3 rasis beginning from mesa, kranti in kala, semi diameter of diurnal circle, carkhandas in asu for Purusottama Ksetra (Puri), rising times of rasi at equator in asu and udayantara phala in asu for convenience of students. Intermediate values can be known by proportional increase. Corrections to Moon 37 ^ Verses 161-162 - Prayer and end - Supreme lord had directed Brahma to create grahas for knowing the earned karma of previous births and fate in the present birth. Brahma, in turn, regulates motion of planets through sighra, manda, pravaha, pata etc. The same supreme Lord Jagannatha has created Purusottama Ksetra for emancipation of beings. I pray that Lord Jagannatha living at NUacala. Thus ends sixth chapter describing kranti, accurate sun and moon etc in Siddhanta Darpana written by Sri Candrasekhara born in a bright royal family of Orissa, for purpose of educating students and for tally in calculation and observation. Chapter - 7 THREE PROBLEMS OF DAILY MOTION (Triprasnadhikara) 1. Scope - There are three problems regarding daily motion of earth, or rather it is used to find their answers - (1) Place - Longitude or Latitude can be determined from daily motion. Both are needed to find the location of a place, specially in sea journey, when there is no other land mark for identification. (2) Direction - North south direction can be measured roughly by a magnetic compass also, which gives other directions also. But this causes a lot of errors, because magnetic north pole is different from geographical north pole, which is on the axis of earth's rotation. In addition, there are local and general magnetic disturbances. Ac- curate method of finding the directions is only by astronomy, whether on land or on sea. (3) Time - Measurment of time intervals are most accurate now with quartz watches for common use and most accurate laser and atomic watches for scientific use. However, that gives average standard time. True or apparent time can be found only by inclination of sun from vertical position. This is related to measurement of longitude also, as simultaneous measuring of time through sun at two places will be different, the difference depend- ing on longitude. Thus time difference or Ion- Three Problems of Daily Motion 351 gitudinal difference can be calculated from each other. Siddhanta Darpana has treated this chapter in briefest manner and one of the vital use i.e. measurement of longitude has been left out. It has been explained roughly for purpose of making desantara correction in madhya graha in chapter 4. One reason of such neglect is that use of astronomy for navigation had ceased for Indians, who had lost the traditional excellence. This doesn't mean that astronomy is not needed for this purpose now. Even in modern astronomy, exactly the same methods are used for finding directions, place and time. With use of telescopes, their accuracy has increased, but formula is same. Another reason for leaving some topics has been stated by the author that many more methods have been explained in detail by Bhaskaracarya, whose book is most popular. Hence they need not be repeated. Before explaining individual methods, it will be useful to give a general idea of various right angled triangles used for calculations. (2) Latitude triangles For calculation of 3 problems, some convenient right angled triangles are formed, whose one of the angles is latitude or aksamsa . Hence they are all called latitude triangles or 'aksaksetra' in Indian astronomy. The other angles of such triangle are obviously 90° - <p, and 90° as it is right angled triangle. 90° - <p is called colatitude or lambamsa. The side facing angle <p is called base (bhuja or bahu), side. Facing 90° - (p is upright (koti) and the side facing right angle is hypotenus (karna). 382 Siddhdnta The radius R of the celestial sphere is assumed tqfc be 3438 or, more correctly 3437'44" (which is value of one radian). (1) Let S be the sun (or any other heavenly body) on the celestial sphere at any given time, SA be the perpendicular dropped from S on the plane of the celestial horizon, SB the perpendicular dropped from S on its rising setting line and AB the perpendicular from A on same line RT. (R is rising point on horizon and T is setting point). OS = R, altitude of S is L SO A = a. Hence height of S = SA = R sin a is the sanku. SB (hypotenus) is called 'Istahrti'. It has beenw called' 'dhrti', 'svadhrti' T Istadhrti', 'nijadhrti' etc. AB is called 'sankuntala' or 'sankvagra' Figure 1 L AS B = <p, hence AASB is a latitude triangle. In this Base upright Hypotenus Sankutala Sanku or R sin a Svadhrti or Istadhrti (1) (2) When S is on prime vertical, SA is called sama - sanku' AB 'agra' and SB 'samadhrti' or tad-dhrti'. Base Upright Hypotenus Agra Samasanku taddhrti (2) (3) When S is on prime vertical, then if a perpendicular AC is dropped from A on taddhrti Three Problems of Daily Motion 3S3 gB two more latitude triangles ACB and ACS are formed, AC = R sin d where <3 is declination. CB is called earth sine (ksitijya), kujya, Bhujya or mahajlva etc.) SC = taddhrti - kujya Base Upright Hypotenus Earth sine R sin d Agra (3) R S in d taddhrti Samasaniku - - - (4) (4) When Sun is on the equator and S its position on the celestial sphere at midday, SA is perpendicular on the plane of celestial horizon and O is centre of the celestial sphere, then SAO is again a latitude triangle. Then L OSA = <p Base Upri^t Hypotenus Rsin0 Rcos0 R (5) (5) When Sun is on the equator, then at midday, the gnomon, (a vertical pillar of 12 unit length called sanku), its shadow (equinoctical midday shadow - palabha, aksabha, palacchaya visuva chaya etc) and hypotenus of the mid day shadow(called palakarna, palasravana, akskarna, aksasruti etc.) also form a latitude triangle. This is called fundamental triangle and has been explained in previous chapter for calculation of lagna, day time etc. Base Upright Hypotenus Palabha gnomon or 12 palakarna - - - (6) Then there are two altitude triangles for sun Base Upright Hypotenus (1) Sanku drgjya or (R sin a) natajya R v) (R sin Z) (2) Gnomon or 12 shadow Hypotenus of (8) shadow '.. -is 384 Siddhanta Darpana (3) When the sun is on the meridian, sanku is called 'madhyasanku' or madhyahna sanku'. Shadow is called 'madhyahna chaya karna'. (4) When the Sun is on the prime meridian (Samamandala), sanku is called sama sanku, shadow is samacchaya and hypotenus of shadow is samacchaya karna. Translation of the text Verse 1 - Scope - For happiness and benefit of the people, I begin this chapter named 'triprasna' which will give knowledge of dig (direction), desa (location) and kala (time) in simple language. Verses 2-5 : Finding the cardinal direction To determine the directions, a place is made plain like a surface of water. It is cleaned and a circle of semidiameter 24 angulas is drawn. At centre a Sanku of 12 angula height is kept. Shadow of sanku will touch the circle twice (when its length is 24 angulas). Both points on circumference are joined by a line and with each point as centre, circles of 25 angula radius is drawn. Both the circles will intersect at two points and common parts of circles between them will form a fish shape. The points are like mouth and tail of the fish. The line joining them will be north south line which will be perpendicular on the line joining shadow position between first and second chaya. Kranti movement is negligible and is ignored. North south line will cut the circumference on two points called north and south points. A perpenducular on that line at the centre will cut the circle in east and west points. For finding ■;-i Three Problems of Daily Motion 3S5 angular directions, arcs between east, north, west and south are bisected. Notes (1) Types of sanku - Bhaskara I in his commentary on Aryabhatiya has described the following views - Some astronomers prescribed a gnomon (sanku), whose one third in bottom is in shape of a prism on square base (caturasra), one third in middle in shape of cow's tail and one third in the top in shape of spear head. Some other have prescribed a square pris- moidal gnomon. The followers of Aryabhata I, used a broad (prthu), massive (guru) and large (dirgha) cylinderi- cal gnomon, made of excellent timber and free from any hole, scar or knot in the body. For getting the shadow ends easily and correctly, the cylinderical gnomon was surmounted by a fine cylinderical iron or r wooden nail fixed vertically at the centre of the .upper end. The nail was taken to be longer thanivthe radius, jof the gnomon, so that its shadow was always seen on the ground. (a) Height of gnomon - Gnomon could be of any length, but its height was divided into 12 units by convention. Smallest was gnomon of 12 angula length, because it was portable and easy to handle, (about 9.8" = 24 cm). Whatever may be length, it was called 12 angula marked by 12 equal division which will be cieariy seen in the shadow. Angula also was divided into 60 pratyahgula for accurate measurement. It may be mentioned that accurate measurements were *7- * <?*■■ -■■ <" 386 Siddhanta Darpi based on very long gnomons. Visnudhvaja 6- Kutubminar at Delhi was one such pillar. Sinci this indicated or marked (like a flag or dhvaja) th position of sun (Visnu) it was called visnu-dhvaja Its arabic translation means the same thing, Kutu means north south direction (Kutub-numa=com pass) manar or minar is measurement or tower fa that purpose. (b) Testing the level of ground - Test prescribed by Bhaskara I, Govinda Sva and Nflakantha is - • • When there is no wind, place a jar of wat on a tripod on the ground which has been mad plane by means of eye or thread, and bore a (fine hole at the bottom of jar, so that water may hav a continuos flow. Where the water falling on th ground spreads in a circle, there the ground is perfect level. Where water accumulates, it is low It doesn't reach at high level. The same principle of 'water level' is used foi modern levelling instrument. A long hollow gla cylinder is filled with water with a small air bubbl in it, when the cylindrical rod, in kept on lev- ground, along the length touching the surface; bubble is at centre. The other side of length to kept on ground is made flat so that it doesn't roll (c) Preparing the ground - Ground should be plastered - so that it is no destroyed by pressure of walking, wind or rainsi A prominently distinct circle was drawn with centr as centre of base of sahku. This line also had 1 permanent or indelible marks by groove gr per*j manent marks. Sankaranarayana (869 AD) tells tha| lines were drawn with sandal paste. This may because sandal was available in his area. Three Problems of Daily Motion 387 Verticality of sanku was tested by means of plumb lines (lambaka) on 4 sides. It seems that fixed length compasses were used for drawing circles. This will be conventient for bigger circles and length of radius will not change in process of drawing. Hence, the radius has always been indicated as fixed. This is not necessary for finding perpendicular bisector. (2) Cardinal directions : Let ENWS (figure 2) be the circle drawn on the ground where gnomon is set. Let Wi be the point where the shadow enters the circle (in the forenoon), and Ei the point where the shadow passes out of the circle (in the afternoon). Join Ei and Wi. Line EiWi is directed from east to west. Figure 2 - Cardinal directions Its perpendicular bisector is found by drawing two arcs of equal radius greater than 1/2 EiWi. This can be any length greater than this, 25 angula radius prescribed here meets the condition. A fish figure is formed with Ni and Si like mouth and tail point of the fish. Since Ei Wi was east west, its perpendicular bisector NiSi will be in north south direction. NiSi being bisector of chord EiWi, it will pass through the centre O and meet the 388 Siddhdnta Darparj^ circle at points at N and S indicating north and south. Line EW parallel to E a Wi through centre O will mark east and west points E and W on the circle. Angle points Ai, A 2/ A 3 and A4 between cardinal directions can be found by bisecting arcs EN, NW, WS and SE. As the sun moves along the ecliptic, its declination (kranti) changes. By the time shadow moves from OWi to OEi, the sun traverses some distance of the ecliptic, and its declination changes (though very small.) Hence, EW is not he true position of east west line. This minute correction was described first by Brahmagupta (628 AD), Bhaskara II (1150), Srfpati (1039 AD) etc. As the correction is very small, this method is good for practical purposes. Figure 3 (3) Correction for Kranti change - From the east west line ew found as above, we make a circle with ew as diameter (Fig 3) Let d = correction in ew for change in kranti. Three Problems of Daily Motion 389 O = latitude of the place <5 = declination of sun when shadow tip enters the circle in forenoon at W <5 = Sun's declination when shadow tip leaves the circle in afternoon at point e. K = chaya karna Then d = K (sin a " Sin W cos <p To apply this correction, a circle with radius d is drawn with e as centre which cuts ew circle at e' towards north when sun's ayana is towards north (as shown in the figure), e' is south from e if sun's ayana is towards south. Now e'w is the correct east west line. This figure is for situation when sun is having south kranti with respect to the place, so that shadow end is in north part. The south kranti will decline in north ward motion of north ayana, hence will be at lesser distance in north direction compared to w. Thus e' is north from e. Derivation of formula : Assuming constant declination, w and e points have equal shadow lengths, hence their directions Ow and Oe are inclined at equal angles from ON direction. It will be proved that K Sin (5/Cos0 is the agra or the distance of shadow from the east west line passing through mid day equinox shadow end. Hence the change in north south position will be difference in the agras at places e and w. Hence this was named agrantara correction by Caturvedacarya and then accepted by Sripati. 390 Siddhanta DarpanaM Unfortunately, derivation of this formula is not possible without use of spherical trigionometry 1 in celestial triangles. Three dimensional diagrams! are difficult to make on paper, they are approximate J indications only. Figure 4 (b) Figure 4 (c) Fig 4 (a) is yamyottara or meridian circle NPZS. (half circle over horizon), SEN is horizon showing south (S), East (E) and north (N) points. Z is vertical and P is pole of equator EQ. Hence ZQES = NP = (p = latitude of the place. In north kranti, sun is moving in a diurnal circle R X V parallel to eequator towards pole P. In south kranti its position will be like R'V. At a position X of sun, its kranti is distance Q' from equator measured along great circle passig though P. Hence PX = 90°- 5. Distance of sun from Z is measured along great circle Z X B = ZX = z. Figure 4B is the direction circle with sanku at O in which WE and NS are direction lines. R is the palabha position on equinox mid day. DD' is east west line through it. At any instant OS is Three Problems of Daily Motion 392 shadow. Its distance from east west line WE is SM called agra jya. Thus agra is the angle a between east horizon E and direction X of sun in a circle through vertical. Thus a = E X arc or L EOX = l_ SOM. In Fig 4(a) it is EB arc. on horizon circle (This direction along polar circle is kranti) Bhuja of chaya = SM = OP = OS Sin a SC = Distance of shadow end from DD', east west line on equinox day = Karna vrttagra. In APZX, Cos (90°- 6) = cos (90°- <p) cos z + sin (9O°-0). sin z cos (90° +a) where, L PZX = 90° + a or, Sin 6 - sin cos Z + cos 0. Sin Z. Sin a. K Multiply both sides by ^, where K is shadow length = V12 2 + s ^ / 12 is sanku and S is shadow. Then K. sin „_ _, ■ T/ p . «. — = K Cos z tan <p + K Sin z. Sin a - cos 6 - - (1) But K Cos z = 12, K sin z = S (2) from figure 4 (c) chaya bhuja b = S Sin a already shown Hence b = K sin z. Sin a (3) __ Ksin<3 ^ ± _ Thus — = 12 tan + b cos 6 But 12 cos (p = palabha = s = equinoctical mid day shadow (OR in fig 4b) .;■-# > A*l /Si! ■^ <■&* .;$ 392 Siddhanta Darparj^ Ksin<5 Hence — = s + b (4) cos 6 When Sun is on horizon, ER is agra A in 4(a). In A PRN (ZPNR = 90°) Cos (90° - d) = Cos <p Cos (90° - A) sin d or Sin A = ....(5) cos <p This agra Jya is in a circle of radius R. Reducing it to circle of radius K it is called Karnagra Ksind a = K Sin A = — cos <p Thus a = s + b (6) In the figure 4(b) Karnagra is difference of s and b i.e. Karnagra SC = PR = PO-RO = s - b Sum or diffrence depends on opposite or same | direction of shadow bhuja and palabha. J ™ , , K (sin d ' - sin d) \ t , 1 Thus the formula ■ — is dif- | cos <p I ference of two shadows in north south directions | by which they should be corrected to make its ends in true east west direction. (4) Alternative methods : Vatesvara, Bhaskara I and II, Lalla etc have given many other methods also, which deserve to be mentioned. (a) Mark the points of extremities of two equal shadows, one before midday and one after that. Line joining them is east west line when . due correction is made for change is sun's kranti. i Three Problems of Daily Motion 393 This is same as the above method. (b) When the sun enters the circle called prime vertical shadow of a sanku is exactly in north-south direction, i.e. smallest shadow. It will be zero, when kranti of sun is same as aksamsa of the place, and not useful. (c) Bhuja and koti of a shadow (its distance from east west or north south line) is calculated. Two bamboo strips equal to bhuja and koti are taken. Koti strip is laid from centre towards west and bhuja strip is laid from shadow and towards south, so that their other ends meet. Then koti will be in east west direction and bhuja in north south. (d) Any heavenly body with zero declination, rises exactly in east and sets exactly in west. (e) The point where star Revati (£ Piscium) or sravana (Altair or a - Aquilae) rises is the east direction. Or it is that point which is midway between the points of rising of citra and svati. Only those stars will rise in east which have zero kranti Observing citra and svati was used by people living in north of,30°N. Sudhakara Dvivedl has written in Digmimamsa, that sravana, whose celestial latitude is about 30° N cannot rise in the east, as it will nevere have 0° kranti (minimum 30 1 1 - 23- = 6-° North Kranti). (e) The junction of two threads which pass through the two fish figures that are constructed with the extremities of three shadows (taken two at a time) as centre is in the south or north relative ^94 Siddhanta Darpanq to the foot of the gnomon, according as the sun is in the northern or southern hemisphere. With the junction of the two threads as centre draw a circle passing through extremities of the three shadows. The tip of shadow of a gnomon does not leave this circle in the same way as a lady born in a noble family does not discard the customs and traditons of the family. Same views had beeen expressed by Lalla Sripati and Bhaskara I (629 AD.) But this has been rightly criticised by Bhaskara II (1150 AD). As the sun is moving on a circle, locus of the line from sun to sanku top will be a cone with sanku top as apex. Its intersection by horizon plane will be always a conic section. The horizon plane is inclined at angle (d + <p ) with sun's direction which is not 90 °, hence it cannot be a circle. As the shadows at sunrise and sunset time are of infinite length, they will be in general a hyperbola extending up to infinity. When (<5 + cp) = 90° which is possible only within polar circle, its locus will be circle. Siddhanta Darpana has mentioned this view in verse 85 of this chapter and has criticised it there and in goladhyaya. However, this method will be approximately correct if the central position of hyperbola i.e. positions near mid day are taken. Verse 6 : Relations be- tween sanku and Chaya - Add the squares of sanku and chaya and take the square root of sum, which will he chhaya karna. Square of sanku (144) is substracted from karna square and square root of the dif- S Ch5y5 Figure 5 OQC nree problems of Daily Motion ( , e nce is chaya which is base or pada. Square root ^difference of squares of karna and chaya xs bhu,a r nku = 12) Line joining ends of bhuja (ianku) and koti (chaya or pada) is called karna. Note : Relation are obvious from figure 5. OV = Sanku = 12 length = Bhuja for the angle z of Sun's direction from vertical (Z.SVO - Z) OS = chaya = Koti or pada for angle z. VS = Line from sanku tip to shadow tip = Karna of chaya. ' OS is in horizontal plane, OV is vertical, hence Z.VOS = 90°. Thus VS 2 = OS 2 + OV 2 Verse 7 : Method to find square root. Steps - (1) Given a number mark the even (sama) places and the odd (visama) places from right (unit place) by horizontal and virtical lines. -1-1 -1 Example 11 97 16 (2) Substract the greatest possible square trom the last odd place. (3) Always divide the even place by twice the square root upto the preceding odd place (4) Substract from the odd place (standing on the right) the square of the quotient (5) Repeat the process as long as there are still digits on the right. Notes : (1) This method was first given by Aryabhata ■M 396 Aryabhata -1 -1-1 119716 (3 Siddhanta DarpanM -A 2X3 = 6) 29 (4 24 57 4 2 2X34 = 68) 411 (6 408 VA = 36 -6 2 X 346 New Method 3 3 11 97 16 (346 9 64 4 297 256 686 4116 4116 X X This is short version of same Aryabhata. method. (2) Proof of Aryabhata method - (1) Put xa = [vTT], xi = 3 11 - X! 2 = 2 (ii) Divide 29 by 2xi with quotient x 2/ x 2 = 29 = 2xix 2 +5 (iii) 57 - x 2 = 41 (iv) Divide 411 by 2 (10 x a + x 2 ) = 2 x 34 411 = 2 (10 Xl + x 2 ) x 3 + 3 (v) 36-x 3 2 = Three Problems of Daily Motion 397 Thus we have 11 = xj 2 +2 29 = 2xix 2 +5 57 = x 2 2 + 41 411 = 2x 3 (10xi + x 2 ) + 3 36 = x 3 2 Multiply these equations in order by 10 4 , 10 3 , 10 2 , 10 1 and add. Corresponding terms are can- celled, as 2 x 10 4 = 20 X 10 3 , 5 X 10 3 = 50 X 10 2 , 41X10 2 = 410 X 10 we get 11X10 4 + 9 X 10 3 + 7xl0 2 + 1x10+6 = xi 2 X 10 4 + 2x! x 2 10 3 + x 2 2 10 2 + 2 xix 3 10 2 + 2x 2 x 3 10 + x 3 2 or 119716 = (xlIO 2 + x 2 .10 + x 3 ) 2 = (3.10 2 + 4.10 + 6) 2 = (346) 2 or V119716 '=■■ 346. Some times we get smaller number at odd place then numbers which will be substracted from that. In previous place quotient is reduced by 1. 1 - 1 - 1 - 1-1 738915489 (2 2X2 = 4 )33 (7 Here quotient should be 8 as 4 X 8 = 32 is less than 33. But at next stage, we w01 get 18 - 8 2 = negative Number. 2X27 = 54 451 -l 2 )33 28 (7 58 -7 2 )99 54 (1 sag Siddhdnta Darpana V73&915489 2X271= 542 )4505 (8 = 27183 4336 This adjustment is to be 1694 done in short method also. -^ 2X2718=5436)16 308 ( 3 16308 09 3 2 Verse 8 : Square root of sexagesimal numbers : Some numbers are expressed in successive divisions of sixty like danda, kala, vikala which are called avayava or components. To find the square root of such numbers, steps are as follows- (1) From the first component i.e. greatest division like danda, we substract the greatest square number. This gives first part of square root in danda (whose square has beeen deducted). (2) If the remainder is less than the square root danda, then it is multiplied by 3. Then it is converted to next lower component (viz kala) and number at that position in kala is added. The sum is divided by square root in danda multiplied by 6 and added with 1. Result will be second i.e. kala component of square root. (3) If 1st remainder is equal or greater than danda root then it is multiplied by 2 and 1 is added. This is converted to 2nd component kala (by multiplying with 60) and number at 2nd component is added. Total remaining kalas are divided by danda root X 4 + 3. Result will be kala component of the square root. Three Problems of Daily Motion 399 Notes : (1) This is a very ingeneous method of finding square root, which I have not come across in any other text. This method of square r0 ot and cube root method in last chapter has not come across the modern world. The method is explained by examples for both cases. Example 1. 7) 60° 20' (7 -7 2 11 -> 11x2+1 = 23° 23x60' +20' = 7x4+3=31)1400(45' 124 160 155 This is more than 7 Thus square root is T 45' Test (7° 45') 2 = / 31 \2 / 961 16 = 60 Which is slightly less than ' the square no. 16 Example 2 7) 50° 20' (7 -7 2 1° 1°X3 = 3° 3°x60'+20' = 7X6+1=43) 200 (4.6 172 280 258 22 This is less than 7° Thus square root is about 7° 4\6 Its square is / 15 60 \ \ '_ 3^ 2 / 40 / 283 x2 / 40 \ = 50° 3' approx. / 400 Siddhanta Darpar^ (2) Justification - This is an approximate method, hence an approximate proof or rather justification of method is given. (i) Suppose A°B' = (a°b') 2 when A-a 2 > a (Example 1) Since A < (a+1) 2 , (a+1) 2 - a 2 > A-a 2 > a or 2a+l > A-a 2 > a __ . ' (2a + 1) + a 3a + 1 Hence, A-a z ~ ~ approx = — - — - This is multiplied by 2 and 1 is added, then, it becomes (3a+l) + 1 = (3a+2)° = (3a+2)60' Now (a+1) 2 > A > a 2 +a = a (a+1) 1 3 1 or A = (a+1) (a+-) = a 2 + - a + - = a 2 + 2ja + ^=(a + f) 2 Hence b = 45' approx (more than half degree) B is betwen V to 59' = 30' on average Hence remainder is (3 a+2) 60'+30' = 180 a + 150 approx Dividing by b = 45 , -= = 4fl + 3.3 approx. Hence the remainder is divided by (4a+3) to get the value of b. (ii)) When A-a 2 < a Since A-a 2 > always, on average we can take Three Problems of Daily Motion 401 A-a 2 = a/2 A°B' ~ A+— * ~ a 2 + - + - 2 ~ 2 2 , 2 2al ' 1 x 7 55 ( a + -^r- + 7i ) + 4 4 2 y 16 1 . 7_ 16 = ^ a + I ) 2 + T^ = ( al / 4 °) 2 = (a° 15') 2 approx Remainder is multiplied by 3 and converted to kala then added to B - 30' becomes | X 3 X 60 + 30 = 90a+30 On division by 15', range of b it gives 6a + 2 Hence it is divided by 6a+l to give approx value of b. Verse 9 : When in astrology, we calculate proportionate life term from value of naksatra/ difference of 1 kala will give age difference of 72 days. Hence component quantity roots should be found carefully. This is a rough method involving some error. Hence it should be checked by squaring. Verse 10 : Multiplication of component numbers - A multiplication of two quantities with 3 components each will be in 9 places. First number is written at the top with three components at 3 places. 2nd number with 3 components is written below, by its first component we multiply the first Hne's components at 3 places. The multiplication b y smaller component is written below it, drifted 1 place towards right. 3rd multiplication by next 402 Siddhanta Darpt tm smaller component is shifted 1 more place towards right. Thus total is in 5 places. First place fronii left is unit (rupa), 2nd place is lipta (1/60 part), 3rd is vilipta (1/60 lipta) and so on. Only 3 placet are taken. Their square root can be found out byg method of verse 9. Otherwise, for accurate calculaj tion, they will be converted to vikala whose squarei root will be in kala. Notes : This method is called go-mutrika in Indian arithmetic. Like urination by cows at separate spots, multiplication is done at different lines. Proceduce is as follows - a° b' c" X d° e' f" ad° db' dc" ea' eb" fa" ec"' fb r " fc"" ad° db'+ dc"+ +ed"' fc"" ea' eb" + +fb"' = A =B' fa" = A° =B' =C" Only A°B 'C" is kept which is sufficient for accuracy. (60 A°+B') X 60 + C" = Vikala Vikala = Kala X Kala (e'xb' = eb" vikala as above) 1 1 1 as — x 60 60 3600 Hence square root of vikala will be in kala. Verses 11-12 : Setting of sariku Circular base of sahku should be plane and from top to bottom, face should be plain and straight (i.e. smooth conical surface). Height of Three Problems of Daily Motion 403 cone and circumference of base will be equal). Shape of sanku may be any type, but l/12th part of its height will be called 1 angula. For finding out time, our own body also can be considered a sanku and the distance of shadow is measured from middle point of the feet. Convenient sanku is of eye level height made of soil or wooden pole. Its centre will be at centre of circle. Radius of base is measured already. Distance of shadow end is measured from base of sanku and radius of base is added to give shadow length. Verse 13 - The shadow meant here is produced by centre of sun. But other parts of sun are not dark and they also contribute to the shadow. Hence the length of the shadow is increased by 1/211 to find the shadow length due to sun's centre. Notes : In figure 6, shadow of Sanku CP due to centre O of sun is CS. Elevation of sun is Z.CPS=z. Due to upper most part X of sun, end portion SS' is also lighted. Hence, only shadow CS' is seen. To find correct shadow, length SS' is added to it. Now PS = CP sec z, CS = CP tan z S'N is perpendicular on SP. Since S'N is very small compared to SP, 404 Siddhanta Darpanmi S' N sun's radius S P sun's distance 1 ■ : 'M = (a known cort*t 219 m stant average value) SP or S'N = — 219 S'S = SN sec z (in right angled triangle S'SN) SP sec z CS sec z 219 or SS' = sin z 1 219 **£m -:-*J31 219 sin z . cos z / or ss \ 1 - SS' \ 109 sin 2z 1 (cs' + ss') CS' 109 sin 2z / or CS' 109sin2z-l Thus the correction will be for less than 1/2 the distance of SS', because shadow is not dark" due to dispersion of light in atmosphere. Logic given here is that correection is equal to sun's ; radius; distance it is not correct. Verses 14-23 : Definitions (Text asks to explain the terms through spherical model constructed of bamboo to imagine the measures correctly. Diagram is a crudle substitue, but without it is impossible to describe). Sanku is called nara or koti also. Chaya is called prabha and bhuja also. Square of bhuja and koti added are square of karna. ■ This koti, bhuja and karna form fundamental triangle. Three Problems of Daily Motion 405 The great circle (straight line for a spherical surface) passing through east west points and zenith (khasvastika) is called east west circle (purvapara vrtta). Earth's equator extended into sky is called celestial equator (Akasa visuva). Its aksamsa is considered zero. Great circle passing through poles and east, west points is called samamandala. Ahoratra vrtta becomes successively smaller as we proceed from equator to meru (pole) A sphere of bamboo or wood should be formed to show celestial equator, ecliptic, eccentric circle of planets and other circles. On any day, if the midday shadow of sanku is north from sanku, then its difference from equinox midday shadow is called agra (more correctly karna vrttagra). If shadow is south from gnomon (sanku) base, then sum of equinox shadow (north for north latitude only) and this shadow is called karna vrttagra. On equinox day sun makes day and night equal while on equator (perpendicular to equator on that day). Thus the distance of sun on this day from svastika of a place is aksamsa or palamsa (angular distance from equator) of the place. Palamsa is the nati (angular distance from zenith or svastika) on equinox midday. Its angular height from horizon is unnatamsa equal to lambamsa (complementary to aksamsa - distance from north pole). 406 Siddhanta Darpcti 12 angula sanku and palabha multiplied by radius (3438) and divided by pala karna give respectively lambajya and aksajya. Notes (1) Figure 7 is as per commentary by Pandita Bapudeva Sastri on surya siddhanta. ■ i! .*;^ E A 1 ti^ K x. P \/l \ \ / R p X. \ °\ 1 X / H y f ,**:# XJ M B 4 ■■*& ■'I ; % '"I N Figure 7 - Definitions in spherical triangles ZANB is yamyottara mandala (meridian) pass- ing through two poles P, P', and zenith Z. All the other circles have been projected on this plane for diagram purpose. Samamandala is great circle through Z, N and east west points. Ksitija (horizon) is circle passing through north south east west points. ACB is its diameter in the figure which is in north south line. Nadi mandala is celestial equator. Its diameter is ECF. P and P' are dhruvas (north and south poles) of earth. PCP' is a diameter of unmandala perpendicular on diameter of nadimandala (or its diameter). GH is diameter of ahoratra vrtta (diurnal circle) of sun (or any planet or star). This meets ■:J. Three Problems of Daily Motion 407 pCP at L (bisected there) and ksitija at O. Let EM be perpendicular to AB. Then EZ is aksamsa and CM is its sine or aksajya. AE is lambamsa and EM is its sine or lambajya. CE is trijya, Thus EMC is a latitude triangle with lambajya, aksajya and trijya as its sides. It is called W. CL is distance between nadi mandala and ahoratra vrtta - and is equal to krantijya. L is point of intersection of ahoratra vrtta and unmandala and LO is perpendicular on line of intersection of ahoratra vrtta and Ksitija (this line is perpendicular to the plane of paper i.e. diagram). This LO is kujya. CO lying on ksitija is the distance between purvapara and udayasta sutra and is agra (both the lines perp. to paper plane). Thus CLO is another latitude triangle with sides as krantijya, kujya and agra - called X. Let the sun be at K. Perpendicular KD to ksitija is also called sanku (or mahasanku). DO is sankutala and KO, istahrti. OKD is another latitude triangle called Y. Midday sanku is called madhyahna sanku. Suppose sun is at E, the equinoctical point, let CR be sanku of 12 angulas. RT is its shadow perpendicular to it meeting ECF in T. RT is called palabha, and CT is pala karna. CRT is the basic latitude triangle called Z. Verses 24-27 : Kranti from Palabha Now I tell the method of finding current declination (angular distance from equator - kranti) °f sun forn palabha (midday shadow) 408 Siddhanta Darpani , . sT.r- Midday shadow on north south line is multiplied by radius (3438) and divided by karna. Arc of this jya is found in kala. This is natamsa of sun (distance from kha-svastika = zenith). If shadow end is south from the equinox mid- day shadow, then sun is having north kranti. Then kranti kala of equinox day (aksamsa) is f added to natamsa (kala) which gives sun's kranti. (for north latitude). Sun's equinox shadow and mid day shadow on desired day being in one direction, difference of kranti and natamsa is taken. They are added when in different direction. According to surya siddhanta, palabha (on equinox day) is found out from aksajya of the» place. Lambajya in found by taking square root of difference of squares of trijya (1,18,19,844) and| aksajya. | Notes H :'i 1 i4i N Three Problems of Daily Motion 409 Let HZPN be the observer's yamyottara mandate and Z be the zenith. Let EQ be the nadl- mandala, HON ksitija and P Dhruva (north). Let S be the sun at mid day (In south declination towards south point H from Z). S will be towards N in north declination. ZS is its natamsa or distance from zentih (vertical). HS its unnatamsa (elevation from horizontal) and SE its kranti (distance from equator - shown north here). ZE is aksamsa. Draw SA perpendicular to ZO. Then AS is natamsajya and OA is unnatamsa jya. Produce ZO to cut the circle at Z'. Cut OB = 12 angula. Draw BC perpendicular to OZ' meeting SO produced at C. Then OB is sahku, BC madhyahna chaya (mid day shadow) and OC chaya karna. Natamsa ^SOZ = ^BOC is given by BC Chaya Sin ABOC = — -^ which is the formula. Now when S and E are on same side of Z, (as in figure), the shadow BC will be in opposite side of both. In this case, SZ = EZ - ES Or Natamsa = Aksamsa - Kranti When S is on other side of Z i.e. at S', the shadow will be in side OZ'H, opposite to equinox shadow. Then, ES' = EZ + ES' Or Kranti = Aksamsa + natamsa For same sides it was Aksamsa - natamsa 410 Siddhdnta Darpana Verses 28-32 : Sun from shadow - Now I tell the method of finding sun's position from shadow. If natamsa and aksamsa are in same direction (i.e. shadow on equinox midday and desired mid day is in same direction from sanku base), then we take the difference of these. When they are in different directions, then we take the sum. This will give kranti of sun (in case of difference, it is in direction of greater quantity, for sum, it is direction of either. Kranti jya is multiplied by trijya (3438) and divided by jya of paramakranti (1370). This will give bhuja jya of sun. Its arc is found in kala. If sayana sun is in first quadrant, this arc itself is position of sayana sun. If it is in 2nd quadrant, it is substracted from 6 rasis, in third quadrant added to 6 rasis. If sayana sun is in last quadrant, arc is subtracted from 12 rasis. Ayanamsa is deducted from this value to get true sun as measured from mesa 0\ Sphuta or true sun is substracted from its mandocca and mandaphala correction is done in reverse manner for madhyama surya. By repeated procedures, madhyama surya will be more accurate. Notes : Calculation of sayana sun involves two steps (i) Finding kranti of sun as described in verse 27. (ii) From kranti of sun to its sayana position, which has been described in chapter 6 verse 96. There the formula has ben used for the reverse process, i.e. to find sun's kranti from position of sayana sun. Three Problems of Daily Motion 411 Kranti jya x Trijya Bhujajya of sun = Parama kranti This formula has been proved there. Now sayana sun is reduced to true sun by reverse process of finding sayana. Earlier ayanamsa had bene added (it may be substracted for periods before 493 AD or after 2200 AD according to book - which is not correct). Hence, it will be substracted now. Madhyama graha from true graha is again a reverse procedure of finding true graha. It has been explained in verse 166 of chapter 5. For sun, only manda correction is done. Verses 33-34 : Shadow from sun's position of midday Sun's position will give its kranti as explained above. Aksamsa of a place is known. If both are in different direction, they are added, to give natamsa of sun (inclination from vertical). If both are in same direction, their difference is taken. (Here direction of aksamsa is opposite to direction of equinox shadow i.e. direction of equator from the place). Thus in north hemisphere, aksamsa is south). Thus we get natamsa at mid day. Its bhujajya and kotijya is calculateed 12 x nat amsa jya ch *y a = K^i 12 x radius (3438) chaya karna = ^p 412 Siddhanta Darpana Figure 9 Note : This is obvious if we consider figure after verse 5 or 13, reproduced here. OV is vertical direction at a place where OA is sanku of length 12. OB is shadow on horizontal plane. Thus ZVAS = Z.BAO = natamsa of sun, Z.BOA = 90° Now chaya BO = OA tan z = U sm z 12 x R Sin z R Cos z Cos z 12 x natamsa jya chaya Karna AB = 12 X radius Kotijya of natamsa OA 12 x R Cosz R cos z Kotijya Verses 35-37; Unmandala sanku Unmandala is great circle passing through east, west points and north and south poles. (Defined in verse 23 - figure 7). Its northern part lies above horizon in north hemisphere places (like India)). Unmandala is horizon of equator, its sanku is formed when sun (or a planet) enters unmandala. Then perpendicular from it to east west line is unmandal sanku. When sun is in north kranti, it Three Problems of Daily Motion 413 rises earlier than equator, thus at unmandala, it has risen at equator horizon and gone above horizon at local place. Palabha x Kranti jya Unmandala sanku = yasti = Pala karna unmandala sanku x trijya Carajya When sun is north from equator, yasti + U. sanku = madhyahna sanku. For sun in south, yasti - U. sanku = M sanku. NoteS- Figure 10 - Unmandala sanku ZSZ'N is the meridian of a place of latitude 3>. S,N. is north south line on horizon. ECE' is diurnal (ahoratra) circle's diameter when sun in on equator. QQ' is diameter of ahoratra when its kranti is d . P,P' are north and south pole, joining line is diameter of the circle passing through east and west points on horizon, so perpendicular to plane of paper like equator circle. 41 * Siddhdnta Darpaq* PCP' is the north south line of equator and unmandala is horizon circle there. C is east point, CR is agra. Perpendicular from planet at unmandal to horizon, is equal to its projection BD in meridian plane. Thus BD is unmandala sanku. On diurnal circle projection, sun moving from Q' above, rises at horizon at point R. At position B it is on horizon of equator and rises there. Thus sunrise is earlier in north hemisphere when sun has north kranti. Half of ahoratra vrtta diameter BQ = Dyujya Difference between equator and horizon rise = BR = Kujya (in kala angles) Difference in rising time in asu = Kala for equator = CA = Carajya EQ = d (Kranti), BC = R sin d= Kranti jya, Aksansa <p = arc S'E or PN or angles BCR etc marked with Zsign. BF and QT are perpendiculars on vertical line CZ. Q is mid day time of sun, so TC = madhyahna sanku = R sin z where z = natamsa QZ = Z.QCZ Thus, madhyahna sanku is TF length more than BD i.e. unmandala sanku. * • TF = yasti (or madhya yasti at madhyahna time) = Height in vertical direction above equator rising point. This height at any other position is- called ista yasti. In latitude ABCD, M Three Problems of Daily Motion 415 BD BD Sin = ^77 = ~ — : 7 BC R sin or, unmandala sanku BD = R sin 6 Sin $ - - -(1) (R sin 6) (R sin 0) x . or ^ — as stated FT = FO + OT = (BO + OQ) Cos<S> = BQ Cos $ But BQ is at angled from equator hence, BQ = R cos d Hence yasti FT = R Cos 6 Cos <S> (2) yasti = carajya _ Unmandala sanku R ■ * by dividing (1) with (2). Here, yasti = madhyahna sanku - Unmandala sanku (4a) when sun kranti is north. In south kranti MM', sanku at B' will be in opposite direction. Then yasti = madhya sanku + unmandala sanku " (4b) Value of carajya in (3) has already been proved in chapter 6. It is proved as in APCA, BR//CA CA BR Hence — = — CP = R, BR = BC tanO = R sin<5 tanO (from diagram) BP = R cos d Hence carajya CA = R tan d tan O used in (3) Siddhdnta Darpana Verse 38 : Alternative method for madhyahna sanku - Madhyahna sanku Unmandala sanku X Antya Carajya Notes (1) Antya = Trijya + cara jya (defined later) = EC + CA = EA (Fig 10) xt BD CT TO + OC Now 1ST = ™ (similar triangles) = ~ BR QR 5 ; QO + OR BD BR CA CT ~ QR ~ EA or Madhyahna sanku CT BD X EA _ Unmandala sanku x antya CA Carajya when sun is having south kranti, Antya = Trijya - Car jya. Trijya in asu is half day length at equator, carajya is difference in half day length at own place. Thus antya in asu is half day length at any place. (2) Yasti is a stick with length equal to trijya = 3438 used to measure vertical height of sun from horizon, as ratio of trijya - hence it gives sine values. Thus, the height measured from the position of equator sunrise is ista yasti. In north kranti, at equator rise time, it "is below horizon, so its vertical height at equator sunset time can be measured, which will be almost equal and opposite. For north kranti it can be measured directly. Hence, the name yasti has been given. Yasti and all sanku measurements are in the direction of local vertical i.e. line passing from Three Problems of Daily Motion 417 earth's centre to the surface point. Heights of sun along this line from equator rise time will give yasti. This gives a measure of equator time i..e udayantara correction. Verses 39-44 - Agra and Kama Vrttagra - Jya of natamsa (R sine of angular distance from zenith is called drgjya and its kotijya (R cosine) is called sanku jya Kranti jya x palakarna Madhyahna agra - 12 (4aftku) " " " < A > Agra at madhyahna is south or north as sun is having north or south kranti. Kama Vrttagra madhya agra x chaya karna = Radius(3438) ( ' (Karna Vrttagra is distance of shadow end at any time in north direction measured from equinox mid day shadow) Alternatively, Madhyagra - *** *» X .*** (A') °^ Lambajya Kranti jya x chaya karna mtx Kama Vrttagra = - T 7-7-z — (B') • & Lambajya Say ana sun in six r a si's starting from mesa is in north hemisphere and in six rasis from tula is in south. When sun is in north and karna vrttagra is more than palabha (equinox mid day shadow), then their difference will be south bhuja or bahu of shadow (bahu is length of shadow in north south direction). Sun in north and palabha more than karna, then their difference will be chaya bhuja in north direction. 418 Siddhdnta When sun is in south, then karna and palabha are always added to get chaya (These rules have been stated for places north hemisphere like India). Notes : ■I V _ ■ I | rj Figure 11 - Karna Vrttagra NZSZ' is meridian, or yamyottara vrtta of a place passing through north horizon point N, south point S and khasvastika (zenith) Z - i.e. vertically up point. NES is horizon circle (east half shown) P = Pole of equator EQ AiR diurnal circle of sun at north kranti (declination) Ri, R 2/ R3 are its three position. Ki, K 2 , K 3 are positions of sun projected on equator through polar circles. KiRi, = K 2 R 2 = K 3 R3 = Kranti of sun (almost equal for a day) PRi, PR 2/ PR3 are polar distances of sun. ZEZ' is sama mandala through east and west points of Three Problems of Daily Motion 419 horizon, zenith (svastika) points. R 2 is sun's position on svastika. Polar great circles from Z to positions of sun meet equator at Ai, A 2/ E and A 3 . Thus natamsa are ZRi, ZR 2/ ZR 3 , angular distance from svastika. A 2 Ri, ER 2/ A 3 R 3 are angular elevations (unnatamsa) EAi, EA 2 , EA 3 are agras of sun. Now in spherical triangle PZRi Cos PZRi - Cos (FRl) ■" Cos (ZRl) x Cos ( pz > sin(ZRi) x sin(PZ) PZRi = 90° - agra (a), PZ = 90° - PN = 90° - <I>, O = aksamsa PRi = PKj-KaR! = 90° -6,6 = Kranti ZRI = z natamsa TT . Sin 6 - Cosz. Sin <£ Hence, sin a = sin z . cos <X> Sin 6 - Cot z . tan O (1) Sin z . cos O _ Palabha ^ 12 But tan O = — — — , Cot z = — S = shadow c . chaya S 5>m.z = . = — , K = chaya karna chaya karna K y „ . sin (5 K 12 palabha Hence, sin a= — . — - — x cosO S S 12 j. S K sin 6 r— - palabha cos <I> r _ . K sin 6 or Sana = r~ - palabha (2) cos <I> r v ' ■ - w. 420 Siddhanta Darpafui S sin a = bhuja of chaya measured in north south direction from base of sanku. Thus, karna vrttagra = bhuja + palabha (By definition) Ksind ,„, x K.V. = — (B') cos <I> as stated earlier Relation (2) holds when sun is having north kranti and is north of samamandala. Then bhuja is in south direction, which may be taken positive. Bhuja (south) = (Karna vrttagra - palabha), when in north kranti, sun is south of sama mandala angle 'a' is negative (north wards from point E is +ve direction). Then - Bhuja = K.V. - palabha When sun is in south kranti, d will be negative, a will be negative so - Bhuja = - KV - palabha When north direction values are taken Bhuja = KV +' Palabha. These are the rules for bhuja of chaya. Here, madhyahna agra or madhyagra has been the name of agrajya at sun rise time which may be named A. Thus A = R sin ao where a is agra at sunrise Then, natamsa Z » 90°, cos Z = and Sin Z = 1, equation (1) becoms Sin <5 Sin a = — cos O Three Problems of Daily Motion 421 A ■ '.' R sind R x R sin<5 or, A = R sin a = - — -— - — — — (A') cos R cos <X> To find (A) and (B) relations, we have Palakarna R ■ 12 ~ Rcos <f> R sin d X Palakarna Hence, A = 12 (A) From (B'), we have K.V. = R sin 6 cos <I> \ K X R madhyagra X chaya karna radius Verses 45-51 : Relations in sama mahdala- When shadow of sanku falls on east west line, then shadow, chaya karna and time (indicated by nata or unnata amsa of sun) .-•' all are in sama mandala i.e east west vertical circle passing through zenith. At this point kranti of sun is equal to aksamsa of the place. When north kranti of sun is more than the aksamsa (for north hemisphere) of the place, shadow is always south of samamandala. Shadow is north of sama mandala when sun's north kranti is less than aksamsa of the northern place or kranti is south. Summary -I'- Shadow on sama mandala - then, Kranti = aksamsa 2. Shadow south ; N. Kranti > aksamsa (north) 3. Shadow north ; N. Kranti < north aksamsa or south kranti 422 Siddhanta Darpann (A) Samamandala chaya karna = Palabha x lambajya jya of north kranti Jya of north aksamsa x 12 Jya of north kranti Palabha X dinardha karna dinardha vrttagra (b) Sama mandala sanku Jya of north kranti x palakama Palabha a (c) Drgjya = Vxrijya 2 - Samamandal sanku 2 drg jy a x 12 (d) Sama mandala chaya = ' — ; t~t\ v ' • ' J Sama mandala sanku Trijya x 12 (E) Sama mandala kama = r - ; — : — \ v f • ■ • Samamandala sanku (f) Say ana sun bhuja jya Sama mandala sanku x Jya of aksamsa Jya of parama kranti (1370) Notes : (1) When sun is in sama mandala (east west circle), sanku, shadow all are in same plane. Then ista kala agra a = O. Thus from equation (2) after verse 44 - Ksiri-d , , , -' q _ _ palabha cos O r palabha x lambajya or chaya karna K = — — . . . .. — (A) 3 • Jya of kranti as lambajya = R cos O, Jya of kranti = R Sind Kranti is north then, only sun can enter samamandala. • m Palabha X Lambajya = aksajya X 12 because palabha = Three Problems of Daily Motion 423 12 R sin 12 tan <D = — - - part 2 of (A) K.V. = cos 3> K sin d cos 4> ^ 11 -1 t^tt Km Sin <5 For madhyahna, KV m = — (KV m and J COS <£ Km are value at madhyahna or dinardha) Lambajya cos <I> Km or . _ = . . = ^T7~ part 3 of (A) Kranti jya sin o KVm r (2) Sama sanku's and kranti Figure 10, in yamyottara plane, indicates position O of sun on samamandala ZCB. ZCB, unmandala and equator all bisect each other on east west points, East point is C here. Sama sanku is perpendicular from sun in samamandala on horizon. It is equal to perp. from O (projection of sun on meridian) to NS, as these are parallel projections. Thus OC = samasahku (height of sun in unmandala) ZO is distance from vertex along diameter hence angular distance Z is given by ZO = R (1 - cos z) = versine Z. OC = R cos Z In ABCO, ZBOC = <fc (latitude or aksamsa) Sin = — — OC But BC is distance of sun from equator or from centre. Angular distance is given by R sin d = =BC 424 Siddhdnta Darpa^ Hence, OC = BC R sin 6 Here Sin <I> = sin O sin <b palabha Hence samasanku OC = palakarna R sin d x palakarna Palabha R cos z = OC = samasanku Drgjya = R sin Z = V R 2 _ R 2 cog 2 z - - (B) = ^Trijya 2 — samasanku 2 ( c ) (3) For other relations, consider figure 12. OZ is part of samamandala with centre at P. PG is a sanku of length 12 at P. SP and SG are chaya karna and chaya of sanku when sun is at O in samamandala * p Natamsa z == arc ZO = L OPR = L SPG = L POC OC is samasanku OR = PC = R sin Z = drgjya In similar A s PSG and OPC; Fig 12 - Samasanku SG SP PG 12 PC OP OC Samasanku Hence, Samamandala chaya SG drgjya x 12 samasanku - - - (D) Three Problems of Daily Motion 425 (E) Samamandala chaya karna SP _ Trijya X 12 samasnku (4) Bhuja of sayana sun - R sin<5 We have R sin (sayana sun) = — — ; , . ... v J Sin (parama kranti) R sin 6 , ,„* , . But samsanku = — ; — — from (B) derivation sin O or R Sin <5 = Sin O X Samasanku Hence R sin (Sayana sun) Sin <£ x Samasanku Sin (parama kranti) Verses 52-62 Kona Sariku • * From sayana sun bhuja obtained above, we can find true and madhya sun as before. Now methods for konasanku are explained, which is calculated through agra etc. Four points midway between east-north, north-west, west-south and south-east are called kona (angle directions). There are two great circles perpendicular to horizon and passing through kona points (one through NE and SW points and other through rest two points). But they are considered 4, one for each kona point. From Surya siddhanta Kranti jya x Trijya Madhya aera = : — : — t~_ J ° Jamba jya madhya agra x ista karna Karna Vrttaera = „ .. . — LJ • ° Trijya When the sun enters one of the kona vrttas, perpendicular from sun on horizon is called kona sanku. Distance of sun from svastika along kona 426 Siddhanta Darpatj n vrtta is natamsa and from horizon, it is unnatamsdl. i Jya of natamsa (R sin Z) or kotijya of unnatamsa [R cos (90°-Z)] is length of kona sanku. Shadow of 12 ahgula sanku, then in opposite direction of kona is called kona chaya, when in north part, kranti of sun is equal to aksamsa, there is no shadow in kona directions. When midday sun has south nata (altitude), then in forenoon, kona sanku is agneya (east south angle) and in forenoon, kona sanku is nairtya (south west angle). When mid day sun has north nata, kona sarikus in forenoon and afternoon are called isana and vayavya. Now chaya and natamsa can be found. (A) Karani = 2 2 2 12 (Trijya /i — agrajya ) _ ^ + palabha 2 ) , Aeraiya x 12 x palabha (B) Aksaphala or phala = 6 iJ . ^ 72 + palabha 2 (C) Mula = VKarani + aksaphala 2 (D) Kona sanku = aksaphala ± Mula (Sum is done when sun is north of east west line samamandala. If sun is south of sama mandala, difference is taken) Drgjya x 12 (E) Kona chaya = Kona sanku ^ „ , , Tri Jya x 12 (F) Kona chaya karna = — ; — : — • ' J * Kona sanku (G) Drgjya or Kona sahkujya Three Problems of Daily Motion 427 =v Trijya 2 - Konasanku 2 Results E to G are quoted from surya siddhanta Notes (1) Equation (1) after verse 44 is _ Sin d - Cos z Sin <E> Sin a = Sin z Cos <l> w here a = agra at any time, d = Kranti of sun, z = natamsa, <I> = aksamsa of the place. Sun is on konasanku, in forenoon, its agra is 45 ° north or 45 "south from east point (according as kranti of sun is more than north aksamsa or less). 1 Sin a = Sin 45° = yy Palabha Again we have sin <p = vi?uva j^a or pala karna ^ = 12 X agrajya Visuva karna • ■ 12 Cos O = Visuva karna • * Then the equation becomes Sin a x Sin z. Cos O = Sin 6 - Cos z. Sin <p 1 12 12 x agra jya or 7j Sin z J^^ = pa lakarna Palabha ~ Cos z — — palakarna (The agra jya on right side is for sunrise time). i or ^2 sin z x 12 = 12 A " Cos z P- (A = Agrajya at sun rise, p = palabha) 428 Siddhanta Darpana 12 : or Sin 2 z = 12 2 A 2 + p 2 cos 2 z - 2 X 12 A X p cos z But R 2 sin 2 z = R 2 - R 2 Cos 2 Z, Hindu system used. So, - X 12 2 (R 2 - R 2 Cos 2 z) = 12 2 A 2 + p 2 R 2 Cos 2 z - 2 X 12 X A X R cos z x p or 12 2 R 2 \ - A 2 = R 2 Cos 2 z [— + p 2 ] - 2p X 12XAXR cosZ Dividing each side by 12 2 /2 + p 2 „, , ■ 2 x 12 x A x p „ R z cos z Z —z — R cos z 12 2 + P J -> R2 12 2 (y - A 2 ) 12 : = + P : Third term is karanT = = N and coefficient of R cos Z in second term is called phala = F, then R 2 cos 2 z - 2 F, R cos Z - N = O or (R cos z - F) 2 = N + F 2 or R cos z = F ± Vi\f + F 2 But R cos Z .= R cos (natarhsa) = R sin (unnatarhsa) = Kona sanku So Kona sariku = VKarani + Phala 2 + phaja Three Problems of Daily Motion 429 Thus F is added for north kranti of sun. (2) As in samasanku, it can be proved easily Drgjya x 12 that, Kona chaya = Kona sanku Trijya x 12 and Kona chaya karna = — j—r, — J ' Kona sanku ■ Drgijya = R cos z = OR = PC PS = Sanku, GS = chaya, PG = chaya karna OC = Kona Sanku, PO = R GS PG PS PC PO gives the result. OC Figure 12 a Verses 63 to 67 : Calculating natamsa - Audayika agra Jya of sayana sun x Jya of Parama Kranti Lambajya (Already proved) From this formula, when bhujajya of sayana sun is 2431 (R sin 45° = 2431) on equinox day, then sun rises and sets on kona circles. When sun is on equator, it is parallel to east west line at all places, hence natamsa of kona circle in forenoon is equal to natamsa of kona circle in afternoon. (This can happen at 45° north latitude). In forenoon, from sayana sun, kranti, chaya agra, time etc are found. Sun position at midday 430 Siddhdnta Darpana is approximate and successive approximation is needed for kona time. * Verses 68-71 : Shadow from time and vice versa - Now, method is explaine to find shadow length, when time is known or vice versa. By this, true positions of planet or lagna can be known at the time of birth, yajna etc. Steps - Natakala is expressed as time or equivalent angle, a planet takes to reach mid day position in forenoon. In afternoon it is time or angle passed from meridian position. Nata kala _ Nata kala in kala (N) ( ' half day " 3 rasi ~ This is different from nata arhsa = z which is angular distance from vertical zenith, it is more than the distance from meridian. (2) Utkrama jya vers N = R (1-cos N) is found. (3) Antya = Trijya ± carajya In north hemisphere, when sun is in north kranti, sum is used. For south kranti of sun, difference is taken (4) Unnata jya = Cos N = Antya - Vers N Cos N is called ista antya also. v , , Cos N x Dyujya (5) cheda = T .. . , called ista hrti Trijya also (6) Mahasahku or sanku R cos Z cheda x lambajya Trijya (7) Drgjya = V Tri j y a2 - sanku 2 Three Problems of Daily Motion 431 Drgjya x 12 Triiya x 12 , , ' , Chaya karna = — ^~tt already found Notes (1) : Formula (6) can be written as Sanku (Antya - vers N) x Dyujya lambajya Trijya Trijya (Trijya ± Carajya - vers N) x Dyujya lambajya Trijya Trijya Trijya - vers N ± carajya . . . = — }J — u — X Dyujya X Lambajya Trijya R cos N ± Carajya = ^ X R cos^ X R cos O R 2 or R cos z = (R Cos N ± Carajya) X Cos d X Cos O d = Kranti, z = natamsa and O = aksamsa This formula is to be proved. (2) Figure 11 after verse 44 may be referred again Natakala - Natakala is the time in which sun or any other star or planet comes to yamyottara (north south vertical circle) in forenoon. In after- noon, it is time lapsed since it had come on yamyottara. These are called purva and pascima nata - incline to east or west. Unnata kala is opposite to natakala i.e. time taken to rise from horizon in forenoon or the time after which the planet will set in west sphere. Unnata kala = 1/2 day time - natakala 432 Siddhdnta Darpanu When polar circles to equator are drawn through position of sun, the arcs on diurnal circle of the planet are proportional to arcs of equator which are proportional to rising time in asu when arc is in kala or minute. Rotation of earth is along equator with fixed speed and time for V rotation = 1 asu,. Thus in figure 11, natakala at Ri, R2, R3 is the time for planet to reach point R of yamyottara. Natakala corresponding to points Ri, R2, R3 all east from yamyottara are angles ZPRi, ZPR 2 , ZPR3 which are proportional to arcs QKi, QK2, QK 3 on equator. Time from E to Q is half day and angle is 90° = 3 rasi Natakala QKi Hence, _ .. . — = ~r— f or point Kl; sun at Ri half day QE r natamsa = -^T7r~ - - - Result (1) 3 rasi (3) For sun at Ri, in spherical triangle ZPRi Cos(ZRi ) - Cos(PZ) x Cos (PRi ) Cos ^ZPRi = Sin (PZ) x Sin (PRi ) or cos (nata kala) Cos z - cos (90° - <I> ) cos (90° -<3) Sin (90° - $) Sin (90° - d) cos z — Sin O Sin d Cos O Cos d Cos z = - tan tand (A) cos O . cos o Now carajya = R tan O tan d (B) Adding (A) and (B), Three Problems of Daily Motion 433 Cos z R Cos (nata) + carajya = — — — — : — 7 v ' u Cos 3> cos o or Sanku = R cos z = [ R cos (nata) + carajya] X Cos <X> cos d In Indian system sin and cos are to be multiplied by R. Results for obtaining chaya and karna have already been proved. Verses 72-75 : Time from shadow. For this, same formula are used in reverse order - Chaya X Trijya Step (1) Drgjya = ^ ^ (2) Maha sanku = VTrijya* - Drgjya 2 (3) Cheda or ista hrti Sanku X Trijya sanku X palakarna Lambajya 12 Cheda X Trijya (4) unnatajya cos N = . . ■ ■ / (5) Nata Utkrama jya = vers N = Antya-Cos N (6) Arc N is found from this. Its value in kala is equal to asu of natakala. Nata— asu divided by 6 gives nata pala. When sun is in forenoon, this is time before noon and in afternoon, it is time after noon. Notes : Methods can be proved in same way, as previous formula. Verses 76-77 - When nata utkramajya is less than 27 kala, there is a separate method. 434 Siddhdnta Darpana Natasu = V Antya 2 - Unnatajya 2 x - (Trijya + antya) Antva Note : Utkrama jya is 29 kala for 2nd khanda of 7-1/2 \ For smaller values (leess than 7° natamsa) this is an approximate method. Verse N = (1-cos N) = N 2 /2 for small N N 2 or, — = Antya - unnatajya For derivation of this approximate formula and to explain the physical significance of terms used at each stage, it is necessary to show diagrams. Natakala has been explained in both circles, yamyottara (meridian circle) in Fig 13a and equator (visuva) circle in figure 13b. In Fig. 13(a), EOE' is diameter of equator, Z T 3< — T^v M p ■I Figure 13 (A) YSmyottara Vrtta Figure 13 (B) Visuva Vrtta QQ' is diameter of ahoratra vrtta (diurnal circle). NS is diameter of horizon in north-south direction. In north kranti, sun comes on horizon at K, hence Three Problems of Daily Motion 435 in 1/2 day QK is increased from QR (6 hours) by RK. QR = semi diameter of diurnal circle = Dyujya, RK = extra length of half day or advance sun rise time = Kujya. The corresponding lengths on equator circle are propositional to time (arc in kala = time in asu). Here OE = Radius of celestial circle = 3438' kala. OC = carajya. Distance of position X from mid day position Q is called nata kala. Correspond- ing nata kala on equator is measured by arc EX'. EX' = vers N as measured from diameter end E.. Length from centre is OX' = Cos N. EC = Antya = distance along meridian diameter from corresponding positions of sunrise and mid-day = EO + OC = Radius + Carajya Ista antya for position X of sun is its distance along meridian diameter between corresponding positions of sunrise and instant position of equator. Ista antya = X'C = CE - EX' = Antya - nata utkramajya QK = Hrti, XK = Ista hrti • • * * Dyujya = R cos O , where <J> is latitude, Corresponding distances on equator and diurnal circle are propositional, hence !?ta hrti = Hrti _ Kujya Dyujya Ista antya antya ~ Carajya " Radius = Cos O - - (1) Now same positions are represented in Fig 13 (B) but in equator circle and projections on it. Projection of P is at O itself. QTQ' = diurnal circle, ET'E = equator circle - half portions above horizon EQE' are shown. For 4^ Siddhanta Darpana positon M, when sun has zero kranti, both circles are one and nata angle N = ZMOT' = arc T'M. Nata utkramajya *■ T'N, Unnatajya or nata kotijya = ON and natajya = MN. When N is small, T'M (approx) = VqM 2 — ON 2 or Nata asu = Vjrijya 2 - Unnatajya 2 In this position antya = Trijya Hence the formula, nata = V&ntya 2 - Unnatjya 2 ^ (Trijya + antya) ' x — —: = ^Trijya 2 - Unnatajya 2 This case is proved. When sun is having north kranti, horizon point on diurnal circle Ki corresponds to horizon point C, on equator; so that OKid and OK 2 C 2 are in one line. Thus horizons are Ki K K 2 and O C C 2 on diurnal and equator circle. Here T'C « Antya, TO = Trijya At Nata n/ position of sun is at X and X' on equator. Arc XT' = LX OT' = N But sun is seen at X making angle 6 at horizon at K. T / K = TQ+T approx. as K is almost in middle of PC. Since angle is small XT = X 0, A = antya Three Problems of Daily Motion 437 However, we are measuring angle from C in formula ^antya 2 -unnatajya 2 = A &■. A0 R + A Hence, N = — x — - — __ . — . - — - 1 (R+A) or, Hence N = V antya 2 _ unnatajya 2 x 2 — a~ (2) Since we are making measurements from R+A distance T'K = —z—* A cos 6 may be more than R. as A > R. Then angle is measured by substracting R from A cos 0, as the jya is same in next quadrant also. Verses 78-80 : Some precautions When nata utkrama jya is more than trijya, we deduct trijya from it and arc of remaining part is taken. It is added to 5400 kala to find nata asu. When nata asu is more than 5400 asu, we deduct 5400 asu and find jya of remaining arc. This added to trijya is nata utkramajya. Nata asu multiplied by savana dina (21,659 asu) and divided by chakra asu (21600) gives suksma natasu. Notes (1) Calculation for 2nd quadrant is same as explained in note (2) after verse 77. (2) We are taking a savana dina as 21600 asu instead of 21659 asu, hence this proportionate correction is done, . Verses 81-84 : Sun from agra and sama sanku. Now I tell the method to find sayana sun from karnagra and samamandala sanku 438 Siddhanta Darpana Karnaera x lambaiya «**»*■ Chayakarna ' ' ' ' < A *> Kranti jya x Trijya Jya (sayana sun) = Jy . rf paramakranti (A 2 ) According to the quadrant of sayana sun, sayana sphuta sun is found. By deducting ayanamsa, sphuta sun is found as before. Alternatively, Trijya x 12 m Samasanku = 7-7: . (Bi) Samasafiku chaya karna Samasanku x aksajya ^ ^ Jya of paramakranti From (Bi), sun is obtained as before. Notes : (1) Formula Ai and A2 have been obtained in verse 40-41 or in 53. (2) Formula Bi and B2 have been given in verse 47-50. Verse 85 : According to ancient scientists, shadow end of the sanku moves on a circular path on a horizontal plane. This is not correct for all places and all times. This will be discussed in goladhyaya. Now we discuss the method to find time in night with help of conjunction of planets and stars. Note : Locus of shadow has been discussed after verse 5. Its formula for radius of circle has been given by Vatesvara and Bhaskara II. This is correct for only central portion of the hyperbola, which is real locus. According to Vatesvara, one formula for (R + agra) (R - agra) diamter of shadow circle is Mid day sahkutala Three Problems of Daily Motion 439 D Figure 14 Diameter of shadow circle + mid day sanku tala. In figure 14, circle ENWS with centre O is the horizon, with east, north, west and south points. A is the point where sun rises, A' is the point where sun sets and M is the foot of perpendicular on horizon from mid day sun. Then circle through A', M and A is locus of shadow, approximately for central portion A'MA. AB, the distance of A from east west line EW, is sun's agra (at rising time). MO = Z m , = R sine of sun's zenith distance at midday. MF, Distance of M from rising setting line AA' is sun's sankutala at mid day. MF = MO+OF = MO + BA = Zm + agra C is centre of circle A'MA. Let OC = x, Then MC 2 = AC 2 (both radius) or (MO+OC) 2 = FA 2 + FC 2 or (Zm -f x) 2 = R 2 - (agra) 2 + (x-agra) 2 where R is radius of circle E NWS. Solving it for x, we get 440 Siddhanta Darpana R 2 - (agra) 2 _ „ 2x = v ° + agra - Zm Zm + agra R 2 - (agra) 2 or 2 (x + Zm) = _ _; 6 _ + Zm v ' Zm + agra _ (R + agra) (R - agra) + ^ ^^^ mid day sankutala This gives the diameter, as x + Zm = radius of shadow circle. Another formula for this diameter is (shadow) 2 - (bhuja) 2 + (bhuja mid day shadow) bhuja mid day shadow This can be proved from same diagram. Verses 86-87 : Lapsed or remaining part of night is found by observing madhya lagna in sky from position of naksatras (position of their stars given in a later chapter). Ayanamsa is added to madhya lagna. From rising times at equator, lapsed part of lagna in the fractional rasi is found. Then remaining rising time for sayana ravi at night in the part rasi is found. These two are added along with rising times of complete rasis between dasama lagna and sayana sun. From this sum, half solar day is substracted. Remainder is the lapsed time in ghati etc of night. Half day added to the sum is the ista time from sun rise. Similarly, remaining part of 10th lagna rasi, lapsed part of sayana sun rasi and complete rasis from 10th lagna to sun (rising times) - all added and half day of sun deducted gives the remaining part of night. Three Problems of Daily Motion 441 Notes : Method of 10th lagna has already been explained in chapter 6. Verses 88-92 : Rising times of naksatras in Oris- sa - Method to find lagna has already been explained from time of day and night. Now for 22° Ayanamsa, rising times of different naksatras in Orissa are stated, by which true madhya lagna can be found in sky. This will be very useful for sky watcheers who can satisfy their curiosisty. At mid day time, that naksatra is in mid sky in which sun is present. 7th rasi of lagna at that time is asta lagna (setting rasi). The lapsed times of lagna rasis are stated according to the naksatra, which has risen in middle sky - starting from sravana. * (22) Sravana - mesa 94 pala (23) Dhanistha - mesa 230 pala (24) Satabhisa - Vrsa 280 pala (25) Purvabhadrapada - mithuna 24 pala (26) Uttarabhadrapada '- mithuna 174 pala (27) Revati - Karka 49 pala (1) AsvinI - Karka 187 pala (2) Bharam - Karka 256 pala (3) Krttika - sirhha.67 pala (4) Rohini - Simha 177 pala (5) Mrgasira - Kanya 2 pala- "(6) Ardra - Kanya 58 pala (7) Punarvasu - Tula 2 pala (8) Pusya - Tula 144 pala (9) Aslesa - Tula 184 pala (10) Magna - Vrscika 32 pala (11) Purva phalgunl - Vrscika 197 pala (12) Uttara phalguni - Vrscika 285 pala (13) Hasta - Dhanu 62 pala (14) Citra - Dhanu 198 pala (15) Svati - Makara 12 pala (16) Visakha - Makara 151 pala (17) Anuradha - Makara 266 pala (18) Jyestha - Kumbha - 67 pala (19) Mula - Kumbha 231 pala (20) Purvasadha - Mina 80 pala (21) Uttarasadha - Mina 442 Siddhdnta Darpana 1*52 pala. From the difference of rising times of these naksatras, time can be found. Verses 93-94 - Conclusion - Bhaskaracarya II has described many types of quantities from bhuja, koti and karna, etc. in Triprasnadhikara chapter of his siddhanta siromani * and has clarified many doubts by questions and answers. This already exists in siddhanta siromani with his own commentary vasana bhasya. hence I am not repeating all due to fear of big size of book. I have described only those topics in detail, which I have verified personally and have separate views. This subject can be understood only through a good grasp of gola (spherical trigonometry) and ganita (mathematical methods). Then derivation of formula will not be difficult. Hence I have not enlarged the bulk of book by writing proofs. Verses 94-95 : Prayer and end - May lord Jagannatha fulfil my ambitions who is rejoicing with Laksml of unsteady eyes and is* residing at Nilacala (Purl) at 276-1/2 yojana north from equator i.e. 19°48' N latitude and 200 yojana east from Indian prime meridian (passing through Ujjain). Thus ends the seventh chapter explaining three questions (Triprasna) along with views of sages; in Siddhanta Darpana written for correspon- dance in calculation and observation, and education of students, by Sri Candra Sekhara, born in famous royal family of Orissa. Three Problems of Daily Motion 443 Appendix to Tripasna dhikara (1) (a) Local time, Standard time and true time : These three are basis of corrections to planet positions, in chapter 2. True time is time cor- responding to nata kala; position of sun. Local mean time is average time of a locality, assuming 24 hours in each day. Standard time is local mean time of a position taken as standard for a country or a time zone. This time differs from Greenwich mean time by exact multiples of half hours. Like standard time of India is local mean time of place 82° 30' east of Green- wich i.e. 5-1/2 hours more than G.M.T. (b) Definitions - Sidereal time - Point of equinox from which sayana position of sun is measured on kranti vrtta (ecliptic) is moving backwards on ecliptic. Position of sun from this point along ecliptic is rasi of sayana sun or longitude. Position of sun along equator is right ascensian. If measured relative to local horizon of earth, position of sun along equator is nata kala or sidereal time, measured from zenith position of sun i.e. 12 hrs noon. Hence right ascension, also is written in hours. (It may be called visuva amsa or hour angle). When motion of equinox is assumed uniform, time measured from it, is uniform sidereal time. From the true position of equinox, it is called true sidreal time. The differnce beetwen them is less than 1/10 seconds and normally ignored. Sidereal time is west wards, because equinox point is moving west wards like sun due to eastward daily motion of earth. It is the time in 444 Siddhanta Darpana hours after the instant equinox point has crossed the meridian (north south vretical circle of a place). Its circle is completed in 24 hours by definition, hence 1 hour movement = 15° (=360° -r 24) Position of planet in hours of right asencion is 15° per hour counted from equinox position along ecliptic. c ecliptic Y Equator Figure 15 (c) Mean time - Mean sun M is a fictitious point which moves along equator with average angular velocity n of actual sun. Since sun completes one rotation in a sidereal year both along ecliptic and along equator, its mean speeds are same in both the circles. Mean sun on ecliptic is Mi and true sun S. - both coincide at perigee or apogee (mandocca). Y is point of intersection of ecliptic and equator. Y M = right ascension of mean sun Y Mi = mean longitude of sun Y S = true longitude of sun Y M = Y Mi = nt after time t. Mean time at any place is called local mean time (LMT) Since it will continuously vary at every place, local mean time of Greenwich is considered standard for the world called Greenwich mean time (G MT) Three Problems of Daily Motion 445 G A B Figure 16 Let A be a place east from G and B another place further east. Longitude difference of A and B is expressed in hours (1 hour = 15°). Let AB = 1 hours = 15 1° If S and S' are local sidereal times at A and B any instant. S' = 1 .+ S because Y will cross meridian at B, 1 hours before meridian of A its west ward motion. Similarly if M and M' are local mean, times at A and B at any instant. M' = 1 + M 1 is same in both formulas because hour angle and mean sun both increase 360° in 24 hours. To avoid inconvenience due to differences in the local times of various places in a country, the local time of a chosen meridian is regarded as standard time. All the places in that country keep this time and not the local time. Thus the standard time of India is exactly 5-1/2 hours ahead of GMT i.e. time of a place 82°30' east of Greenwich. In very large countris like Russia or USA, the country is divided into zones, each having a differnt standard time. For further convenience, the stand- ard times of these time zones differ from GMT by an integral number of hours or half hours. Hour angle measured at Greenwich from 12 hours noon time is called Greeenwich mean astronomical time. (GMAT) and measured from to 24 hours. Meantime reckoned from mean mid 446 Siddhanta Darpana night at Greenwich is called Greenwich civil time (GCT), GMT or universal time (UT). This also is measured from to 24 hours. GMT = GMAT + 12 h Same is for other places also. 1. (d) Mean and Sidereal conversion In one solar year (tropical), sun crosses Y again after one circle. It takes K = 365.2422 mean solar days, i.e. K revolutions of earth with respect to sun. Hence there are K + 1 revolutions of earth with respect to Y or any star. Thus K+l sidereal days = K mean solar days K+l sidreal hours = K mean solar hours etc. 1 Sidereal days = 1 + — mean solar day K. + 1 = 23 56 4.1 S mean solar units. Mean solar day =1+— sidereal days = 24 h 3 m A. 56.5s sidereal hour etc. 1 (e) Years : Sidereal year is time taken by sun for one complete revolution with respect to stars on ecliptic which are fixed. Tropical year is average interval between two successive returns of sun to the first point of Aries (Y). As Y moves backwards in about 26,000 years, tropical year is slightly shorter. (a) Tropical year = 365.2422 mean solar days (a) Sidereal year = 365.2564 mean solar days Three Problems of Daily Motion 447 ecliptic e (2) (a) Equation of time : From the watches we get mean solar time only and we can get the local mean time after longitude correc- tion from standard time. To. know the true solar time or apparent time we have to add some correction (+ ve or - ve) called equation of time. Let E = Equation of time. a = Right ascension of sun (distance from Y along quator = Y D) 6 = true longitude of sun = Y S along ecliptic. 1 = mean longitude of sun. Then, by definition, E = West hour angle of sun - West hour angle of the mean sum = (S-a) - (S - RA. of mean sun) = (S-a) - (S-l) = 1-a This can be written as ■ E = - (a-0) - (0-1) Here - (a-0) is called the equation of time due to obliquity, because if equator is not oblique a = measured along any circle and this term a-0 = Similarly, - (6-1) is called the equation of time due to eccentricity. In the spherical triangle Y SD of figure 16, L D = 90°, so Cos e = tan a cot 448 Siddhanta Darpana where e is angle beetween equator and ecliptic or tan a = cos e tan (0-a+a) Expanding this by Taylor's theorem and neglecting higher powers of e and 0-a, tan a = (1-—) [tan a + (0-a) sec 2 a] e 2 or tan a=tan a + (0-a) Sec 2 a - -y tan a, approx. £ 2 £ i.e. 0-a = -j Sin a cos a = -j- Sin 2a, Since a = 1 nearly, we can write £ 2 - a = ~r sin 21 which is the required value of - (a-#). Often a-0 is called the reduction to the equator, because a-0 added to ecliptic co-ordinate reduces it to equatorial coordinate. Again $ = v+D where D is position of perigee and v is true amomaly, true position of sun measured from perigee. 1 = m + D where m is mean anomaly. So, 0-1 = v-m = 2 e sin m, nearly. Thus E = 1/4 e 2 Sin 2 1 - 2 e sin (1-D) where E, e and e (eccentricity) all are measured is radians giving numerical values. Expressing it in minutes. E = 9 m . 9 sin 21 - 7 m 7 sin (1+78°) 2. (b) A more accurate value-Put y = tan 2 — Then Three Problems of Daily Motion 449 1-y COS £ = ~ — — 1+y i-y and so tan a = - — - tan 6 1+y e 2ix - 1 Using exponential formula, tan x = — ^ where e = base of natural logarithm, we have e 2ia - 1 _ 1-y e^fl - 1 e 2ia + 1 " 1 + y * ex 2i 6 + 1 or J* = elid + y - e 2 ^(l + ye- 2 ^) 1 + ye 2 '" " 1 + ye 2i * Taking logarithms this gives 2 i a = 2i0 + (ye-™ 1/2 yV 4 * + 1/3 y 3 e ^) - [ye 2 * - 1/2 y 2 e 4 * + 1/3 y3 e 6i * ) = 2i 6 - 2i (y sin 2 0) - 1/2 y 2 Sin40 +1/3 y 3 Sin 60) or 0-a = y sin 2 (9 - 1/2 y 2 sin 4 + | y 3 Sin 60 (1) Again - 1 = v-m 5 = 2 e sin M + - e 2 sin 2M + 4 = 2e Sin (1-D)) + 5/4 e 2 Sin 2 (1-D) + - - - (Introduction to chapter 6) Eliminating 0, we get E = 1-a == tan 2 | Sin 2 1 - 2 e Sin (1-D) + 4e tan 2 1/2 e Sin (1-D) Cos 21 - 5/4 e 2 Sin 2 (1-D) - 1/2 tan 4 £ Sin 41 450 Siddhanta Darpatia The equation of time vanishes four times in a year. E = 9™ 9 Sin 21 - 7™ 7 Sin (1+78°) If we draw sine curves f or y == 9 m . 9 Sin 21 and y =7.7 m sin (1 + 78°) and subs tract one ordinate from the other, we get the graph of E, From gr.aph is can be seen that it vanishes four times around 23 march, 22 June, 22 September, 22 December. Sin 2 1 attains max numerical values of 1 four times in a year and is alternately positive and negative at three times. Hence first term twice has value + 9.9 minutes and twice - 9.9 minutes atternately negative and positive. Thus E is alternately positve and negative, because second term is smaller numerically. Hence E is zero four times a year from theory of equations. (3) (a) Parallax : At any instant, the moon has slightly different directions as seen from different places on the earth. Sun's direction changes much less with the change in position of the observer, because sun is more distant. In case of stars, which are far more distant, the difference in their directions as seen from different places of the earth is too small to be measured. But seen from different places in the earth's orbit, (i.e. at different times of the year), the change in the direction of the comparatively nearer stars is measurable. The change in the direction of a celestial body as seen from different positions is called parallax. For calculation of sun, moon and planets, we choose earth's centre as the standard position (origin of coordinate axis) from which distances are Three Problems of Daily Motion 451 calculated. Due to observation from surface of earth, there is parallax error, called geocentric parallax. For calculation of star position, sun's centre is the standard position and difference in direction due to measurement from different positions of earth's orbit, is called stellar parallax. As geocentric parallax depends upon the distance of the observer from earth's ceentre, we begin by considering the shape of the earth. 3 (b) Shape of the earth - Surface of earth deteremined by ocean level is called the geoid, heights of places above mean sea level being negligible. It is an oblate spheroid i.e. rotation of ellipse along its minor axis coinciding with polar axis of the earth. Semi major axis a of the generating ellipse (equatorial radius) is 3963.95 miles and the semi minor axis b is 3950.01 miles. The fraction (a-b)/a is called the compression; Eccentricity of this ellipse e = 0.082., compression = 1/297. Let C be the centre of the earth, O the observer at any place on its surface, OZ the normal at O to the surface and OZ' the direc- tion which produced back- p . ThIn S n? aSS ^ ?° U S h C < angKrveLl inen OZ is the direction of the astronomical Zenith, OZ' that of the geocentric zenith. Angle between these directions ZOZ' is called the angle of the vertical indicated by V. If <I> and 4>' are the angles made by normals NOZ and line COZ' from centre with major axis- <& = geographical latitude of O 452 Siddhdnta Darpana <I>' = geocentric latitude of O v = O- O' If ellipse is referred to C at origin, it is x 2 y 2 — + — = 1 a 2 b 2 a 2 y Then, Tan O = -^- - - (1) b x and tan <p'- — tan O a ^ tan O - tan <E> ' Thus, tan v = tan (<$-<!>') = 1 + tan <X> tan <I> ' (a 2 - b 2 ^ tan <S> _ (a 2 - b 2 ) sin <S> cos <l> a 2 + b 2 tan 2 <I> a 2 cos 2 + b 2 sin 2 <I> (a 2 - b 2 ) sin 2 O m sin 2 O a 2 + b 2 + (a 2 - b 2 ) cos 2 <f> 1 +m cos 2 * a 2 -b 2 where m = — which is small a 2 + b 2 1 + i tan v 1 + m (cos 20 + t sin 20) 1 — i tan v 1 + m (cos lsp — i sin Ixp 2iv I + me or, e = W 1 + mc" 2 ^ Taking logarithms 2 iv = log (1 + me 2 ** ) - log (1+me" 2 * ) - 1 i = me 2 ^ - - m 2 e 4i * + — (me -2 ^ m 2 e" 4 ^ +-} Hence, v = m sin 2<p - 1/2 m 2 Sin 40 + ~ m 3 Sin 6 3 ^ Distance of the observer O from centre C is indicated by p Three Problems of Daily Motion 453 V* _ y/b _ 1__ a cos f " b sin " ^ (a 2 cos 2 <1> + b 2 sin 2 <&) a cos + b sin <p So /o 2 = x 2 + y 2 = 2 2 2 2 a cos <p + b sin a 2 [1 - (2e 2 - e 4 ) Sin 2 <D] 1 - e 2 sin 2 ^> on writing b 2 = a 2 (1-e 2 ) and simplifying Figure 18 Geocentric parallax 3 (c) Geocentric parallax in zenith distancee In figure 18, let C be centre of earth, O is observer and M the centre of moon (or sun or planet), Let CO = p, CM = r If Z' is a point on CO produced, apparent zenith distance t! of M is ZZ'OM and true (i.e. geocentric) zenith distance zo of M is ZZ'cM. Hence z' - zo = parallax in zenith distance = ZOMC = p From plane triangle OCM Sin p = (pk) Sin z' (1) Maximum value of parallax p is when z' = 90% it is called horizontal parallax p n of M at O. Sin p n = pit If O is at equator, then the parallax is biggest as p has highest value a, equatorial radius. The horizontal parallax at equator Po is 454 Siddhdnta Darpana Sin Po = - r When moon (or the sun) is at its mean distance, r G from earth, mean equatorial horizontal parallax P is, Sin P = a/r D For parallax, earth can be considered almost a sphere then, astronomical and geocentric zeniths coincide, z'=z, p = constant = a. We take r = r approx, then approximate value of parallax is p = P sin z - - - (2) Since z' > zo, moon, sun or planet is distanced away from zenith by distance P sin z approx due to geocentric parallax. This is also called diurnal parallax as it goes through a complete cycle of change through a day. Parallax is maximum when moon or sun rise on horizon, reduce to zero, when on zenith and again become maximum when they set in west horizon. 3 (d) Distance and size of moon is calculated by parallax method only. Oi and O2, places on same meridian are chosen. Apparent zenith distances of M are zi' = LZ\ OiM z 2 ' = LZ-l 2 M Z2' Figure 19 Moon's Distance pi and p 2 are parallax angles Oi MC and O2 MC, when C is centre of earth Three Problems of Daily Motion 455 If CA is in the plane of equator, ZOiC 2 = L Oi CA + Z0 2 CA = Oi + <J> 2 where Oi and <E>2 are geocentric latitudes oi Oi and O2 Then z'i + z' 2 =^Oi CM + pi+Z0 2 CM + p; = 4>i + 4> 2 + pi + p 2 Thus pi + p 2 = z'i + z' 2 - (Oi + 2 ) = M 1 ! Because all values on right side are known, 6 is known. Sin pi = (pi/r) Sin z'i (2) Sin p2 = ipiir) Sin z'2 (3) Eliminating pi and p2 from the three equa- tions, we can know moon's distance r = OM It is more convenient to find value of p2 firsl and then calculate r. From (1) and (2) Sin 6 cos p2 - Cos 9 Sin p2 = (pi/r) Sin zi' or Sin & cos p2 = sin z'2 cos + — Sin z' r r r Eliminating r between this and equation (3), we get p2 Sin z'2 Sin 6 tan p 2 = — — ; — : — pz Sin z 2 cos + p\ sin z 1 This gives p2 and then (3) gives r. Figure 20 Moon's diameter 456 Siddhanta Darpana In figure 20, let moon's observed angular semi-diameter be S and let its linear radius be R miles. If the distancee of moon is r miles as determined above, Sin S = R/r from which R can be determined. In India, parallax in zenith distance is called 'nati' and parallax in longitude is called 'lambana'. Lambana can be measured along equator or along ecliptic. Parallax calculation of moon and sun is necessary for calculation of solar eclipse. 3 (e) Lunar parallax along equator and kranti- Mean equatorial horizontal parallax of moon is 57' and for sun it is 8".80 i.e. 1/388.6 of moon's parallax. Hence, accuracy is needed only in calculation of moon's parallax. Figure 21 Parallax in natakala and kranti D 11 In fig. 21, M and M' ar true and apparent (due to parallax) positions of Moon (or sun) MM' = ^ Sin z' r where, p = distance of obsrever from centre of earth, r = distance of moon from centre of earth and, z' = geocentric zenith distance Z'M. Three Problems of Daily Motion 457 Right ascension and kranti of M and M' are a„ d and a'(5'. Let H and AH be their hour angles (nata kala). MD is perpendicular to PM' and L MM'D = V Small A MM'D can be taken as a plane triangle, so 9 MD MM sin r\ Aa=a -a=-AH=- sin PM sin PM £ sin z' . sin rj r ' cos d By sine formula in A Z' PM', Sin rj Sin (z' + MM') = cos <I> Sin (H+AH) Hence A a = p/r Sin z' Cos O Sin (H+ AH) X cosec (z'+MM') / Cos 6 or A a = p/r cos <J> Sin H sec<5 - - (1) neglecting small quantities of second order Similarly Ad = 6' - 6 = - M'D =- MM' cos rj = pit Sin z' cos r\ From cosine formula in A P'ZM Sin (z'+MM') cos rj = Cos d' Sin <p' ~ sin d .cos 4>' cos (H + AH)} Substituting this value of cos rj and neglecting small quantities of second order A 5 = - pit (cos <5 sin <£'- sin d cos <J>' cos H) - (2) Regarding earth as a sphere of radius a, we can write p and <£' instead of a and <f>. Similarly parallax in longitude (along ecliptic) and latitude (sara) can be calculated by considering P as pole of the ecliptic. Then great circle through P and Z' will cut the ecliptic at T called 'tribhona' lagna as it is 90° less than the rising point of 458 Siddhanta Darpana ecliptic on horizon or lagna. Hence T = Lagna -90°. If t is distance between Z and ecliptic (at T), then it is sara of z or declination of T (tribhona). PZ' = 90° -t then. In stead of nata kala H we take distance of moon from tribhona i.e. v and p is latitude in stead of kranti . Then (1) becomees, A 1 = lambana Al = - p/r cos t Sin v. Sec p (3) At eclipse time, P = almost and sec p = 1. Equation (2) becomes &fi = pit (cos p. Sin t - Sin /S. cos t. cos v) — (4) At eclipse time /? = (almost), so cos p = 1, Sin p = A/3=-p/rsint. (4a) (f) Stellar parallax : In figure 22, let X' be a star, S the Sun and E the earth. Let EX be parallel to SX'. Then EX is the true direction of the star, viz its direction as seen from the sun and EX' is the apparent direction, viz, the direction of X' as seen by the observer on the earth. The difference between these directions is the angle X'EX which is equal to the angle SX'E. Figure 22 - Stellar purallax Three Problems of Daily Motion 459 Let L SEX' = 0' L SEX = SE = a and SX' = d Then from the triangle EX'S in which ZEX'S = 0-0', we have Sin {0-0') = (a/d) Sin (9 (1) Let a/d = Sin f] : then II is called the star's parallax (helio centric or annual parallax). Neglect- ing second and higher powers of the small quantities 0—0' and II, (1) becomes 0-0' = II sin which gives the displacement of the star due to parallax. EX, EX' and ES are in same plane, so X, XI and S are on the same great circle in celestial sphere of the observer. Thus the displacement of star XX' on sphere = f] Sin XS (S is direction of sun on sphere). Parallax in longitude and latitude - In figure 23, X is true position of star in celestial sphere, as seen from Sun at S. X' is its apparent position affected by -£35^ parallax as seen from earth (ceentre of the Figure 23 sphere). Longitude and latitude of stellar parallax MM' is ecliptic and K its pole. M, M' are the points on ecliptic at which KX and KX' cut. XD is perpendicular from X on KM' Let L X' X D = W Parallax is II, 460 Siddhanta Darpana X,p andA',/T, are longitude and latitude of X and X', then A A = X - A = X D Sec£ = XX' cosW seep = n sin x s. cos *P sec p = II sin MS Sec p from the A XMS, in which Z.X = 90° - *V i.e. AA = II Sin (0-A) Sec £ (2) where is the longitude of the sun. Similarly, kp = P' - p = - X'D = - XX sin W = - II Sin XS. Sin W = - II Sin p cos (0 - A ) (3) by a PPl)^ n g sme cosmic formula to AKXS Parallactic eclipse : If we take X, true position of star as origin, XK as the y-axis, where K is pole of ecliptic and XD (perpendicular to XK) as x-axis, the coordinates (x,y) of the apparent position X' of the star are given by x = XD = II Sin (0 - A) and y = - X'D = - II Sin p. cos (0-A) Eliminating 0, we see that locus of (x,y) is the ellipse x 2 y 2 + _, * . . = i n 2 ' n 2 sin 2 £ During the course of a year, the star appears to describe this ellipse, which is known as the parallactic eellipse. If M, M' are taken as positions of X, X 1 on equator and T is position of sun on equator, then right ascension and declination can be similarly calculated. Three Problems of Daily Motion 461 A a = II (cos a cos e Sin 6 - Sin a 6 cos 6) Seed Ad = II (Cosd Sin e Sin0 - Cos a Sin<5 Cos 6 - Sin a Sin 6 cos e. sin 0). The parallax is used to measure stellar distances. Star is seen from the two positions of earth six months away (i.e. 180° away in its orbit). Direction of a star is seen with respect to a far i.e. faint star. Nearest star has parallax of only 0".76 corresponding to a distance of 93,000,000 - °- 76 x " ^ 2 55 60 x 60 x 180 X 10 13 miles This is used to define steller distance in units of parsec which is the distance for which stellar parallax will be 1". Another unit is light year, which is the distance travelled by light in 1 year at speed of 1,86,000 miles/sec 1 par sec = 19 X 10 12 miles.' 1 light year = 6 X 10 12 Stellar parallax is not used in siddhanta texts, but have been indicated only to show the other kind of parallax. Only in goladhyaya it has been mentioned (also in discussion of slghra paridhi in chapter 51 that stars are 360 times the distance of sun. This distance is much more and its parallax is no way connected to change of slghra paridhis in different quadrants. (4) (a) Refraction : The apparent direction of any planet or star changes due to bending of rays coming from that on earth due to refraction in its 462 Siddhanta Darpana atmosphere. This is called 'Valana' in siddhanta astronomy and is calculated empirically. Effect of parallax (nati in kranti or lambana in longitude) is to shift the planet away from zenith. But due to refraction (valana), the planet appears higher i.e. closer to zenith. Both are maximum at horizon and zero for zenith. It is difficult to make exact calculation on the basis of refraction rules, even according to modern theories of physics. We obtain some formula after some simplifying assumptions about variations in density and refractive index of different layers of atmosphere. In siddhanta books, calculations are based on practical observations and the correction is assumed to vary according to natajya as in parallax. According to modern electromagentic theory, refraction of light is due to its reduction of speed, when it enters a material medium from vacuum. Since it is an electromagnetic wave, its speed is reduced due to dielectric properties of the medium, which has effect like resistance. The reduction in speed is more in denser medium. Ratios of speeds is called refractive index. Speed of light in vaccum — *- e = u - Refractive Speed in dense medium index. Since speed of light is maximum in vacuum, [i is always greater than 1. When it comes from a lighter medium to material of higher deensity, then also its speed is reduced Speed in medium A u\ L , c — — — = constant Speed of light in medium B j"2 Three Problems of Daily Motion 463 li, fi\ t and fi 2 are constants for the mediums and increase with their density. fi\ and //2 are refractive index of mediums A, B. N Figure 24 - Plane refraction Due to wave nature of light, a ray AB entering a denser medium at B, bends towards normal NNf to the boundary surface DE. If its angle of incidence with normal is 6 and angle of refraction <b then (figure 24) sin$ = a sin <b ■ fii This is a constant depending only on the optical properties of the two media. Figure 25 Cassini refraction 4. (b) Atmosphere assumed homogenous shell- This is called Cassini's hypothesis and is 464 Siddhdnta Darpana simplest assumption. In figure 24, let O be the observer on the earth, A a star (or planet) and APO a ray which reaches O after refraction at P on the upper surface of the atmosphere. Let ju, be the refractive index of the atmosphere. Then the angles being as marked in the figure. Sin 6 = ft Sin 4> (1) But from the plane triangle OPC, if radius of earth is a, and the height of the atmosphere is h, so that CO = a, CP = a+h, we have Sin 6 = fi Sin O sin t sin O — r = - - (2) a+h a v ' Refraction R = .6 - O - - (3) To eliminate 6 and O, from (1) and (3) Sin (R+3>) = fi Sin <&► or approximately, for small R, Sin R=R, Cos R = 1 R cos O + Sin O = ju Sin O Therefore R = (^-1) tan O (fi -1) a sin g = [(a +h) 2 - a 2 sin 2 £f* by (2) {}i -I) sin £ . = [cos 2 5 +2 ( h/ a) r a pp roximatel y = (a -1) tan £ [1 + <a/ a) sec 2 5] " 1/2 approximately = (a - 1) tan HI - 0/a) sec 2 5] = (ju - 1) tan 5 [1 " 0/a) (1 + tan 2 £)] which is of the form R = A tan £ + B tan 3 £ Three Problems of Daily Motion 465 The simple formula R = K tan £ is true for values of £ not exceeding about 45 \ this formula is true for values upto 75°. 4. (c) Concentric layers of varying density : This assumption also gives the same formula, by an approximate method. Figure 26 Concentric layers of varying density Suppose that any layer of the atmosphere is bounded by concentric spherical surfaces AB, A'B' and that PQR is a portion of a ray of light which finally reeaches the observer O on the surface of the earth. Let C = Centre of earth, CQ = r, CR=r+ Ar Then the normal at Q to the surface AB is CQ. The angles and refractive indices are as marked in the figure. From the laws of refraction M sin ® = (ju + Ap) sin ^ - .. (1) From plane ACRQ sin (0 + A <l>) _ sin*P r r + Ar 466 Siddhdnta Darpana Eliminating ¥, we get fi r sin<D = (a + A/*) (r + Ar) sin (<D + A4>) As this relation is true for any two consecutive layers, ^ r sin O has the same value for every layer. But on surface of earth, r = a (radius of earth) * = £ (apparent zenith distance) fi = ju , (say), depending on density and temperature of atmosphere, so (jl r sin O = ^0 a sin £ (2) Amount of refraction at Q (say AR) = O - W so (1) gives. fi sin <S> = (w + A//) sin (O - A R) = (fi + Ap ) (sin <J> - AR cos 3>) approximately. = 11 sin + Aa sin O - A R. fi cos O so, AR = (Aw/a) tan4> Eliminating O with help of (2), we have ddi sin £ A ,.v pi (r 2 // 2 - a 2 ^g sin z Q V2 To solve the differential equation (4), we assume r - = 1 + s a Where s is small, because the earth's atmos- phere extends only to a comparatively small distance from earth's surface. Putting this in (4) and integrating, we get. R = aao sin £ J^ fit' 1 a" 1 (ju - & sin 2 £ + 2s ft 2 ) Vl dfi Three Problems of Daily Motion 467 ■d Cf*o -2/2 2-2 ^-V2 or R = J 1 n (ju -ju sin £) Is fi 2 1 + fi 2 - fio 2 sin 2 £ d/4 neglecting higher powers of S. It is assumed that z is sufficiently less than 90° to ensure that the denominator fi 2 - po 1 sin 2 £ is not very small or R =,io . sin £ £ — j 2 , 2 ^ /* (a - fi sin 5) . o Sft d/J n"o sm£j — . ... (5) 1 (w z - /^6 sin 2 Q To integrate first term, wee put 1/Ja = t then it is sin -1 Qio sin £) — £ i.e. sin _1 [ (1+x) Sin £] - £ , putting To expand the first tern by Maclaurin's theorem, let f(x) = Sin a [(1+x) Sin £] Then f'(x) = , Sm ^ Vl - (1 + x) 2 sin 2 5 Thus f (O) = Sin 1 (Sin £) = £ c- 3nd f (0) = V - sin 2 6 = tan C Thus the first term in (5) is equal to x tan £ approximately, neglecting higher powers of x. Second term in (5), has a small quantity s as a factor. So its coefficient is changed slightly. Putting // = fio = 1 in it, the term becomes " C T7^7 J sd P 468 Siddhdnta Darpana - sing rfl0 cos 3 C Now by Gladstone and Dale's law fi - 1 + cp where p is the density of the layer with refractive index //, and c is a constant. This gives dft = cdp If p is density of the air at surface. of the earth, the second term becomes - sin £ cos 3 £ Integrating by parts, and supposing that s = when p - O, this becomes i COS 3 L, The intgrated part vanishes at both limits (p = at one limit and s = o at other). The remaining integral is equal to mass of a column of air of unit cross section, extending from surface of the earth to the point P = O. It is, therefore, a constant and can be written as B tan £ sec 2 £ , where B is a constant. Thus R = (a - 1) tan £ + B tan £ (1+tan 2 £) which is of the form R = A tan£ + B tan 3 £ Bradley's formula : He assumed r jU n+1 = constant Also fir sin <p - constant - from equation (2) Therefore, by division sin £ rS " C TT^T h P ds Three Problems of Daily Motion 46$ V n . - = const. - ^ - (6) sin O v ' By logarithmic differentiation n d0 - = cos <D . — *- From equation (3), — = - tan 4> From these two equations dR = (1/n) d<& Integrating from the surface of the earth (where r=a, fi = fi Q and $> = £) to the upper boundary of the atmosphere (where fi = l, r=r' and O' = O' assumed) we get R = 1/n (£-<&')- - - (7) From (6), ^ S 1 i.e. sin 0'= sin f sin O sin £ >o n Then (7) becomes, R = -g- sin -1 (sin g) ] n n This is known as Simpson's formula This can be written as sin £ sTn(e-nR) = M ° n or sin £ - sin g - n R) = /«, » - 1 sin £ + sin (£ + nR) /Lt n + 1 1 #o n - 1 1 or tan - nR = ^ tan (£ - - nR) Writing 1/2 nR for tan 1/2 nR we get 4jo Siddhanta Darpana R = ^f| tan (^ nR) n (jMo + 1) L This is Bradley's formula. 4. (d) Determination of constants - In figure 27, let Xi and X 2 be true positions of a circumpolar star at its upper and lower culminations (positions on meridian). Then Figure 27 P Xi = PX 2 = 90° - 6 , PZ = 90° - <& Therefore, ZXi = (90°- <p ) - (90° -<3) = d - <p ZX 2 = 90° - tf>+ 90° - 6 = 180° - <t>- d Hence ZXx + ZX 2 = 180° - 2<p (1) If the apparent zenith distances at upper and lower culminations are £ and ?- then ZXi + ZX 2 = Z£ + 2£ (la) Z£ = £ -f A tan £ + B tan 3 £ Z £' = ?' + A tan?' ' + B tan 3 £ Putting this value in (1) we get one equation in 5 and - £'. Equation of two more such stars will be used to determine A,B and <I> . Numerical values of A and B for a pressure of 30" of murcury and temperature of 50° F (or 10 °c) are 58". 294 and - 0". 0668. Three Problems of Daily Motion 471 For values of £ greater than 75 ° , special tables are used based on observations. The refraction when a body is in the horizon is called the horizontal refraction, and its value is about 35'. From equation it will be « for £ = 90° as tan 90* = oo, hence equation is not correct for such values. H^— Z p w Y Figure 28 (e) Refraction in visuva amsa and Kranti - In figure28, let X be the true position of a star and X its apparnt position as affected by refraction. Then ZX' X is a great circle and XX' = K tan £ where £ is the apparent zenith distance ZX . Let the hour angle (natarhsa) and kranti (declination) of X be H and d for X' these be H' and d\ Join PX; PX' and produce them to meet the equator in A and B. Draw X'D perpendicular to PM. Then, since XX' is small, the trianlge XX' D may be regarded as a plane triangle. Now the correction to be added to the apparent right ascension a' to obtaint the true right ascension (Visuvarhsa) a is a-a'. But a-a' = - AB = - X' D sec X'D (as X' D is almost equal to arc X' D with centre P) * 472 Siddhanta Darpana = - X'X Sin rj Sec <3' = - K tan £ sin rj sec 6' rj is given in APZX, by sine relation sin (90° - O) _ sinH sin rj sin £ as PZ =90°- <p , ZX = £, so, sin ?/ = Sin£ cos Sin H Similarly the correction to be applied (added) to 6' is <5-<5' But d -6' = - DX = - XX' cos rj = - K tan £ . cos?7 4. (f) Effect of refraction on sun rise and sun- set Hour angle (natamsa) H of sun's centre when rising is (Figure 29) CosH = - tan 0tan d (1) where <p is latitude of the place and 6 is declination (kranti). Let H + AH be the natamsa of true sun when the apparent sun is rising. At this instant, the true sun is really 35' below the horizon, its true zenith distance being 90° 35'. Hence, from the AP S'Z cos (90° 35') = Sin <p Sin 6 + cos <p cos d cos(H + AH) or, - sin 35' = Sin <p Sin d + cos <f> . cos 6 (cosH - AH. sin H) nearly or — Sin 35' = - AH. Sin H. cos <p cos 6 by (1) Three Problems of Daily Motion 473 N Figure 29 This will give the advance time of sun rise H in radian, it will be divided by sin V to get the value in asu. (1) is obtained from equation for natakala cos H. cos O . cos 6 = cos z - sin <p. sin 6 At sun rise time, z = 90°, cos z = Thus apparent day length is increased and if sun rise at parama kranti time is measured, it gives a higher value of parama kranti. This may be one of the reasons for assuming its value as 24° instead of 23° 27'. 4. (g) Shape of sun's disc at sunrise or sunset - Lower limb of the sun is at a greater zenith distance than the upper. Hence due to refraction, the lower limb is raised more than the upper. Thus the sun appears flattened. This effect is maximum when sun is near the horizon. Let S be sun's centre, a its radius and P any point on sun's limb, (figure 30) Let ZS = z and let PQ be the perpendicular from P on ZS. On account of refraction, let P be displaced to P' and let P'Q' be the perpendicular from P' 474 Siddhdnta Darpana on ZS. Then, since QP is small, the zenith distances of Pand Q are the same. So PQ will be displaced to P'Q'. Now take SZ as the X axis and perpendicular to it through S as the y axis. Then if the coordinate of P' are (x,y), we have x = SQ' = a cos ^ + QQ' = a cosW + K tan (z-a cos *P) = a cos + K (tan z-a cos V sec 2 z) (1) x-axis Y axis Figure 30 - Sun disc at rising time PP' = K tan z where K = \x - 1 Its component along PQ is K tan z cos ¥ (2) But from right angled triangle ZPQ cos W = tan PQ. cot z. Hence resolved part of refraction in PQ dirction is K tan PQ = K.PQ, since PQ is small Three Problems of Daily Motion 475 y = P'Q' = PQ - K. PQ = (1-K). -PQ = a (1-K) Sin^ - - - (3) Eliminating V from (1) and (3) , we see that the apparent figure of the sun is the ellipse (x - k tan z) 2 y 2 _ j (a - ak sec 2 z) 2 + a 2 (1 - k) 2 Thus sun appears elliptical at sunrise and sunset. Chapter - 8 LUNAR ECLIPSE Candragrahana Varnana Verse 1 : According to views of smartta, vedic, purana knowers, there are unlimited good results from auspicious works at the time of grahana (eclipse) like bath, homa, charities etc. People repose faith on tithi calculations after seeing eclipse as predicted. Due to this importance, eclipse (solar and lunar) is described now. Notes : (1) This chapter describes the general methods applicable both to solar and lunar eclipse. Calculation of solar eclipse needs some special methods, which will be discussed in next chapter, named surya grahana. (2) Auspicious effects of grahana are subject of 3rd part of Jyotisa called samhita and need not be discussed here. However, calculation of grahana is a very complicated process. If such a rare event occurs as predicted by calculations, it is an excellant proof of correctness of theories and formulas. Verses 2-6 : Possibility of eclipse. Lunar eclipse - At the ending time of Purnima (when moon-sun = 180° exactly), difference of moon with rahu and ketu is calculated. When this differncee is less than 13°, then lunar eclipse is possible. Lunar Eclipse 477 Solar eclipse - Similarly, at the end of amavasya (when moon - sun = 0°), moon and its pata (rami or ketu) are calculated. Difference of moon from any of the pata being less than 18 *, solar eclipse is probable. We calculate amanta time (when sun=moon), from earth's centre. Pascima nata of candra X 1/3 is substracted from this time and we again correct the true moon at this corrected amanta time. Again we calculate, vitribha (tenth lagna) for this time. 1/60 of its natajya is added to second true moon of this time, when moon and nata are in same direction. We substract, when they are in different directions. If this is less than 34 then, solar eclipse is probable. Sometimes, when south nati (in meridian circle) is less than 1°30' then solar eclipse is probable. When drgvrtta is kranti vrtta, then difference of candra and its pata being less than 7°, solar eclipse is possible. Notes : (1) Reason of eclipse - When moon passes into the earth's shadow, it fails to receive light from sun. This causes an eclipse of moon. This can happen only when the sun and moon are on opposite sides of earth, i.e. on full moon time (Purnima when moon-sun = 180°.) M1 Figure 1 - Lumar eclipse 478 Siddhanta Darpana Let S be the centre of the sun, E of earth. The cone touching sun and earth has its vertex at V. Then the portion of cone from earth upto V is the shadow cone of earth called umbra (bhubha). This is completely dark as no light from sun reaches in that portion. Another cone is formed by tangents in transverse direction with vertex in opposite direc- tion between earth and sun. The portion of this cone after earth and beyond umbra (shadow) is partly dark and called penumbra (avatamasa). Mi, M4, are points on moon orbit at boundary of penumbra, M2, M3 on boundary of umbra. Between Mi M2 or M3 M4 portion, brilliance of moon is reduced, which are described as colour of eclipse but no eclipse is formed. In portion of orbit M2 M3 completely within the shadow cone of earth (bhubha), there is an eclipse. At point 1, moon's disc just starts contact with, shadow, this is called first contact or 'sparsa' (touch) kala. At point 2 moon's disc just enters completely in the shadow called second contact or 'nirmlana' or 'sammilana' (closing the eyes). When complete eclipse is aboui to end i.e. moon's disc starts coming out of shadow at point 3, it is called third contact or 'Unmllana' (opening the eyes.) At point 4, moon completely comes out of shadow. It is called fourth contact or moksa kala (freedom time.) When the moon is not completely eclipsed, the times of maximum eclipsed portion correspond to 2nd and 3rd contacts. Lunar Eclipse 479 D Figure 2 - Solar Eclipre (2) Reasons of solar eclipse : An eclipse of sun is caused by moon coming in between the observer and the sun. If the whole of sun is hidden behind the moon, we have a total eclipse. If moon covers only part of suns disc, we have a partial eclipse. When apparent diameter of moon is smaller than sun in a total eclipse, the eclipsed part of sun is surrounded by visible circle of sun, it is called annular eclipse. These are called 'sarvagrasa, . and, 'khanda grasa' or kankana grahana respec- tively. This can happen only on amavasya, when sun and moon are in same direction. In figure 2, if observer is anywhere inside the shadow cone of moon AVE, the whole of sun is hidden from his view. If he is in the extended cone FVG, only the central part of sun is hidden by the moon. If the observer is within penumbra CAV or VAD (except FVG portion), he will see a partial eclipse of sun. It can be seen that at point O in extended shadow cone only the inner portion BB' of sun is obstructed. In this case, moon is smaller, so its shadow cone doesn't reach earth's surface. In this eclipse also, sparsa or first contact is time when eclipse starts. 'Nimilana' is time when maximum eclipse starts (or total eclipse) i.e. 2nd contact'. UnmHana or 3rd contact is when maximum lS otb' A 4S0 Siddhanta Darpana or total eclipse is about to reduce. 'Moksa', or 4th contact is time when sun is completely visible. (3) Why eclipse doesn't occur on every purnima or amavasya?- The inclina- tion of moon's orbit to the ecliptic N is about 5°. Hence the maximum dis- . c t Figure 3 - Lunar eclipse not occurring tance of moons a centre from the ecliptic is 5°. Now the axis of the earth's shadow lies in the plane of the ecliptic. Moon's diamater is about 1/2° and diameter of earth's shadow at distance of moon is about 1-1/2°. So moon will touch the shadow, when its centre is at a distance from centre of shadow by less than 1/2 (1/2° + 1-1/2°) = 1° approx. Thus, for most of the time, moon passes clear out of the shadow. Eclipse is possible only when moon is near N, the point of intersection of its orbit with ecliptic. The northern point of intersection, from where orbit goes north of ecliptic is called rahu and other southern pata is caled ketu. Hence, rahu and ketu are said to cause eclipse. For solar eclipse also, sun and moon should be in the plane of ecliptic, so that moon's shadow touches the earth. Thus on every amavasya, when moon and sun are in same direction from earth, solar eclipse doesn't occur. Shadow of moon is almost a point when its shadow cone touches the earth or it may not touch at all. Thus its radius may be taken as zero, at distance of earth (from moon). Earth's radius makes an angle of about 1° Lunar Eclipse 481 at moon. Hence as distance between shadow centre and earth centre less than 1°, solar eclipse is possible. Thus within similar distance of moon from its node, solar eclipse happens. In solar eclipse, sun is not covered, it is only locally obstructed, like obstruction of a cloud. Away from shadow cone at a short distance, sun is visible because parallax shift of moon is 57' compared to 8". 8 of sun, which is not obstructed there. K a Sr R-a "^N ^ y c i 1 VE# M * V Figure 4 - Earth's shadow in moon's orbit (4) Size of earth's shadow in moon's orbit. S and E are centres of sun and earth V is vertex of shadow (umbral) cone of earth.* FA is one of generators of cone and v its semi vertical angle. Let moon touch the umbral cone at N and NM be perpendicular to EV. Then s, the angle subtended by NM at E, is the angular radius of earth's shadow at the distance of moon. Z.ENA = Pi = horizontal parallax of moon approximately as AE is almost perpendicular to EM. S = sun's angular semidiamter, P = horizontal parallax of the sun = a/r. a = radius of earth 482 Siddhanta Darpana r = distance of S from E. (sun from earth). R = radius of sun. Then s = Pi - v from ENV of which Pi is an exterior angle. = Pi - Z.KES, if KEI IAF = Pi-KS/SE nearly as SF is almost perp. to SE = P! - (R-a) It = P + Pi - S. or s = P + Pi - S This gives the theoretical value of s, but it is found that actual observations give the value 2%. larger, because earth's atmosphere absorbs light. Angular radius of the penumbra at the distance of moon can be shown similarly to be P+Pi+S (S is angular semidiameter). Approximate value of radius of shadow is about 42' after adding 2% for atmosphere. It varies with change in distance of sun and moon from earth. As moon moves 360° in 29-1/2 days with respect to sun, i.e. with respect to shadow, it will be fully in shadow till it covers (diameter of shadow - diameter of moon) = 2X42' - 30' = 54' approx. The time in covereing the distance. x — x 24 hours - \— hours approx. 60x360 2 4 This is the maximum duration of a total lunar eclipse. (5) Ecliptic limits of Moon— Figure 5 is celestial sphere part for observer. N is node of moon's orbit. C is centre of earth's shadow on M Lunar Eclipse 483 ecliptic. M is centre of moon. Moon's orbit meets ecliptic at N which is its node. Angle betwen the orbits is i. In the diagram moon is just touching shadow. If C was Q when M was at N, then NCi is called the lunar ecliptic limit. If Figure 5 . Ecliptic Limit of moon shadow is nearer then moon will definitely pass through the shadow. If C is away, moon cannot touch it and there will be no eclipse. Since M moves about 13 times faster, only moon's motion is being discussed. As sun is diametrically opposite to O and other node of moon's orbit is opposite to Ni lunar ecliptic limit is also the distance of sun's centre from nearer node of moon's orbit at the instant, moon is crossing the ecliptic. Let NC = x when moon is crossing the ecliptic. Let n, ni be the angular velocities of the sun and the moon (radian per hour) in planes of their orbits. Let the time counted* from moon's centre passing through node be t hours. Then at time t, NC = x + nt, NM = n a t Taking NCM as a plane triangle, CM 2 = (x+nt) 2 + (n^t 2 ) - Irnt (x+nt) cos i- (1) CM is a minimum when t is given by 2 (x+nt) n + 2ni 2 t - 2ni x cos i - 4 nint cos'i = by differentiating equation (1) with respect to t. 484 Siddhanta Darpana Substituting the value of t given by this in (1) and simplifying, minimum value of CM is x ni sin i (n 2 + n? - 2nm cosi) V2 " ® When moon just grazes the earth's shadow in its course along its orbit, the minimum value of CM - must be equal to the sum of the radii of shadow and moon. Hence (2) is equated with | <P+ Pl -S) + Sl where S and Si are angular semi diameters of sun and moon, P and Pi are equatorial horizontal parallax of sun and moon. As all the quantities P, Pi, S and Si are variable, the lunar ecliptic limit also varies. Its greatest value, called the superior ecliptic limit is 12°.l and the least value, called the inferior ecliptic limit is 9°. 5 These limits are for a partial eclipse. By equating (2) to the difference of radii of the shadow and the moon, we can find limits for a total lunar eclipse. (6) Commencement of solar eclipse When partial eclipse of sun starts, the transverse common tanquent BA touching sun and M B . (W C Figure 6 - Start of solar Eclipse Lunar Eclipse 485 moon at B and A respectively just touches earth somewhere, say at C. Let a, b and R be the linear radii of earth, moon and the sun. ES = r, EM = n and L MEC > 0, ZMES = x Then r cos (0 + x) + R = a (1) ri Cos 6 = a +b - - - (2) Divide (1) by r and (2) by ri and substract. We get cos - cos (0+x) =■ — .+■ — + — or 2 sin — sin 2 / \ x 6+ 2 \ J ri ri a_ ^ a R ri ri r r As x is small and is nearly 90°, this gives, approximately, x = Pi + Si - P+S Figure 7 - solar ecliptic limit Solar Eclipstic Limits - The solar ecliptic limit is the distance of the sun from the nearer node of the moon's orbit, at the moment of new moon, if a solar eclipse is just possible on this occasion. Let MN be moon's orbit and SN the ecliptic, so that N is a node of the moon's orbit and let the inclination of the moon's orbit to the ecliptic be i. = ZMNS. 4S6 Siddhanta Darpana Let M and S be centres of moon and sun at the instant of a new moon occuring when the sun is near N. By the definition of a new moon (amavasya), longitudes (rasi) of M and S are the same. Let P be the latitude of moon when at M. Let M', S' be the positions of the moon and the sun t hours later, and MSN is taken as a plane triangle. Let MM' = x Then change in moon's longitude in t hours is x cos i. Then change in the sun's longitude in t hours, i.e. SS' is m x cos i, where r ate of change of sun's longitude m " rate of change of moon's longitude Then S'N = SN-SS' = p cot i - mx cosi and M' N = P cosec i - x If M'S' = D, we have D 2 = (P cot i-mx cos i) 2 + (P cosec i -x) 2 - 2 cos i cos i - mx cos i) (P cosec i-x) - (1) Only variable in this is x. Differentiating it with respect to x, minimum value of D is given by (p cot i-mx cos i) (-m cos i) - (p cosec i - x) - cos i {-fi cot i - m ft cos i cosec i+2 mx cosi) = p sin i or x = 1 - 2 m cos 2 i + m 2 cos 2 i Substitution in (1) shows that the smallest value of D is Lunar Eclipse 487 (1 - m) ft cos i (1 - 2m cos 2 i + m 2 cos 2 if 1 When numerical values of m and i are substituted, it is seen that the value of this expression is very nearly ft cosi, i.e. the value after supposing m = (i.e. very small speed of sun). Putting, therefore, ft cosi = S +Si + Pi - P We have the condition that the sun just misses being eclipsed. This gives ft = (S + Si + Pi - P) Sec i as critical value of ft within which ft should be for an eclipse to be seen in some part of earth. Solar ecliptic limit is the corresponding value of SN = (S+Si+Pi-P) cosec i Its greatest value is 18°. 5 i.e. the superior solar ecliptic limit; its least value is 15°. 4 which is the inferior ecliptic limit of sun. Thus the text mentions only the superior ecliptic limits of sun and moon as 18° and 13°. If distance of sun from the node is more than this; eclipse is impossible, if it is less than the lower ecliptic limit 15°. 4 or 9°. 5 eclipse is certain at new moon or purnima. If distance of sun is within inferior and superior ecliptic limits on new or full moon, solar or lunar eclipse may or may not happen. Further checking should be done at the ending times of purnima or amavasya by lambana (parallax) of sun and moon and their true diameters and speed. 488 Siddhanta Darpana (7) The other condition of solar eclipse is for a local place. The solar eclipse may happen, but it will be visible for only a small belt on earth's surface through which moon's shadow cone passes. When sun and moon are in same direction from earth's centre, the eclipse will be visible from a place where difference in parallax of moon and sun is less than the sum of their semi diameters (= 34') approx. O' S Figure 8 - parallax in amavasya time Figure 8 shows the position of true amavasya, when moon M and Sun S are in same direction from earth's centre E. When observer is at O in this line, i.e. when moon and sun are on zenith, then the same position remains. When observer is at O', moon is ahead of sun towards east by p = L O'SE of parallax. Thus moon will be in same direction with sun slightly before true position, at true time it goes ahead. Thus for east nata amavasya time is before true time and in west nata it will be after true amavasya time. Surya siddhanta has assumed horizontal paral- lax as l/15th of the daily motion of a planet, on assumption that the (speed X distance) for the planet is constant. Linear speed of every planet is Lunar Eclipse 489 assumed to be same and it comes ot to be (15 X radius of earth) as explained in 2nd part of this book. For moon this gives correct parallax but gives great error for other planets, due to wrong assumption of distances. Maximum parallax are compared below in vikala. Planet Bhaskara II Modern value Minimum Maximum Sun 236.5 8.7 '9.0 Moon 3162.3 3186 3720 Mars 125.7 3.5 16.9 Budha 982.1 6.4 14.4 Guru 20.0 1.4 2.1 Sukra 384.5 5.0 31.4 Sani 8.0 0.8 1.0 Siddhanta darpana has corrected the values for sun and moon (through still assuming same linear speeds) Horizontal parallax for moon = 3388". 22 Horizontal parallax for Sun 31.63 Change in sun's parallax is an improvement caused due to taking higher value of sun's diameter as mentioned in Atharva veda. But still it is about 3.6 times the true value. Changed formula for parallax are Daily speed Sun max parallax = Moon max parallax = 164 Daily speed 14 Thus the parallax of moon is the distance travelled by it in 4 ghatl (60 ghati in a day/15) 490 Siddhanta Darpana according to surya siddhanta and in 4/17 ghati according to this book. For rough calculation, appendix 3(e) after chapter 7 gives the formula (3) as Al = - " cos t sin v r p/r = max. parallax, v is distance from 'Tribhona' lagna, which is taken as zenith as first step. Then the correction in ghati is 4.28 X cos t. X Sin H where H = nata kala For 45° nata (middle position between meridian and west horizon), sin H = 1/ V2, H = 15/2 = 7.5 ghati. This positive correction for pascima nata will be 2.5 ghati or 1/3 of nata kala if t = taken 30° (nata of tribhona) Parallax in sara = p/x sin t. Parama nati = 1760 approximately, hence 1/60 of natajya of vitribha or tribhona lagna is added for calculating sara difference of moon. Assuming nil sara at eclipse' time, this can be maximum of 34' for an eclipse to be possible at that place. (8) Other condition for solar eclipse - When sun is moving on east west vertical line, its kranri being equal to latitude of the place, its difference with moon when apparent longitudes are equal is the north south difference i.e. nati (parallax in sara or latitude). When it is less ,than 1/2 (sum of diameters of sun and moon) or 1°30' then only solar eclipse can happen. Lunar Eclipse 491 When Difference of moon and its pata is less then 7° then also solar eclipse can happen. This is same as 1°30' difference from ecliptic. (9) Greatest and least number of eclipses in a year - Ecliptic limits are as follows - Superior Inferior Lunar ecliptic limits 12 \1 9°. 5 Solar ecliptic limits 18°. 5 15°. 4 1 Lunar month = 29.5 days So, time from full moon to next new moon = 14.75 days. Node of moon moves backwards, making one * revolution in about 19 years. Hence sun makes one complete revolution with respect to node in 346.6 days. Thus, with respect to node, sun moves 360° x 14.75 , ^o o ■ i. ir i 4-u or about 15 .3 in half a lunar month. 346.6 Figure 9 - No of eclipse (A) Least number of eclipses - Figure 9 shows the ecliptic and N, N' are nodes of moon's orbit. Let NSi = NS2 = N'S3 = N'S 4 . = inferior solar ecliptic limit i.e 15°. 4. 492 Siddhanta Darpana Let NM 2 = NM 2 = N'M 3 = N'M4 = inferior lunar ecliptic limit i.e. 9°. 4 Inferior limits have been chosen to find the most infavourable cases in which no eclipse occurs beyond these limits. Movement of sun is in direction of arrow. Si S2 = 2X15°. 4 i.e. 30° 8 but sun moves with respect to N by 2X15°. 3 between two consecutive new moons. Thus in travel from Si to S 2 at least one solar eclipse is bound to occur, because there will be a definite new moon in 30.6 days and sun will be within limit of eclipse. Suppose now that the eclipse occurs when sun is near N, then the sun will be outside NMi and NM2 at previous and next full moons (i.e. 15°. 3 away) while NMi = NM 2 = 9.5 only. Hence, there will be no lunar eclipse in previous or coming full moons. Thus there will be only one eclipse (solar) while sun crosses N. Sun will be at N' after about 1/2 X 346.6 = 173*.. 3 days after it has crossed N. Now 6 lunar months occupy 6X29.5 = 177 days. Therefore, about 4 days after the sun at N'„ there will be a new moon. Then sun is only 3.7 X 3607346.6 =3.84 from N' i.e. will within ecliptic limit of N'S 4 . Thus there will be a solar eclipse. The preceding and succeding full moon occur out side M 3 M 4 as sun moves about 1° in a day. N1M3 = 9°. 5 but N'S = 14.75 - 3.84 = 10.91 on previous full moon. In next full moon N'S = 14.75 + 3.84 = 18.59. Thus there are A no lunar eclipses then. If the year began shortly after the sun had crossed S 4 , the year will end 365.25-346.6 days after Lunar Eclipse 4^3 the same point relative to nodes, so the year will have ended much before sun comes near N again. Hence in such circumstances, there will be only two eclipses in the year, both solar.. (B) Greatest number of eclipses in an year - Now in figure 9, let NM a = NM 2 = N'M 3 = N'M4 = 12 \1 i.e superior lunar ecliptic limit. NSi = NS 2 = N'S 3 = N'S 4 = 18\5, the superior solar ecliptic limit. Suppose further that there is a new moon as soon as the sun enters SiN. There will be a solar eclipse then. Time counted from the eclipse is indicated by H = half lunar month.. Then we haye to examine solar eclipses at new moons at time 0, 2H, 4H, 6H - - - Similarly lunar eclipses are examined on full moons at times H, 3H, 5H (i) There is already a solar eclipse at t = (ii) At t = H, Sun is at 15°. 3 from Si and within MiN at full moon, so there will be a lunar eclipse then. (iii) At t = 2H, sun has advanced 2X15°. 3 from Si; so it is within NS 2 and there will be a solar eclipse. At t = 3H, 4H, HH, the sun will be within S 2 and S 3 i.e. outside all the ecliptic limits, and there will be no eclipses. (iv) At t = 12 H, sun will have advanced 12X15°. 3 i.e. about 184° from Si i.e. 4° from S 3 . So the sun is within S 3 N' and there will be a solar eclipse. 4 9 4 Siddhanta Darpana (v) At t = 13H, sun will have advanced 4° + 15\3 from S 3 , so it is 19°. 3-18°. 5 = 0\8 from N' in N'M4 and there will be a lunar eclipse. (vi) At t = 14 H, sun will be 0.8 + 15\3 = 16°. 1 from N' i.e. will within N'S 4 = 18°. 5. So there will be a solar eclipse. At t = 15 H, 16 H, 23 H, the sun will be between S 4 and Si, i.e. outside all ecliptic limits, and there will be no eclipses. (vii) At t = 24H, sun will have advanced 2X4° = 8° from Si, so it is within SiN and there will be a solar eclipse. (viii) At t = 25 H, the sun will have advanced 8°+15°.3 from Si, so it is within NM 2 , and there will be a lunar eclipse. But this eighth eclipse occurs 14.75 X 25 days i e. 368.75 days after the first eclipse, i.e. about a year and 3-1/2 days after the first. So out of 8 eclipses, 1st solar or 8 th lunar eclipse has to be ommitted in a year. Thus in a year there can be maximum of 5 solar+2 lunar or 4 solar + 3 lunar eclipses depending upon when the year began. (10) Eclipse cycle : In Chaldea, before 400 BC, (may be in time of Sargon in 2350 BC approx,) a cycle was discovered after which eclipses were repeated. This was called Saros cycle of 18 years 10.5 days or 223 synodic lunar months. 223 synodic months = 6585.321 days 242 dracontic months = 6585.357 days =19 X 346.62005 days (Dracontic year) Draconitic year is revolution of sun with respect to lunar node and draconitic month is Lunar Eclipse 435 revolution of sun with respect to its node. Nodes of moon were called Dragons. Visvamitra had mentioned half cycle in Rkveda of 3339 tithis = 111 synodic months + 9 tithis. Example of the cycle for least no. of eclipses in given below - (No lunar eclipse + 2 solar eclipses) Years 7915 1933 _1951 "1922 1940 2958 "1926 1944 1962 Feb. 14 Feb. 24 March 7 March 28 April 7 April 19 Jan. 14 Jan. 25 Feb. 5 Dates of solar eclipse Aug. 10 Ann. Total Aug. 21 Sept. 1 Sept. 21 Oct. 1 Oct. 12 July 9" July 20 All annulus Total Annular July 31 Cycle of years of maximum eclipse Years Lunar Eclipse Solar Eclipses 1917 Jan 8, July 4, Dec 28 Jan 23, Jun 19, July 19, Dec 14 1935 Jan 19, July 16, (Jan 8) Feb 3, June 30, July 30, Dec 25 1953 Jan 29, July 26, (Jan 19) Feb 14, July 11, Aug 4, gan5), next year next year Total Total Total Part Part Part Annular Actual determination of eclipse, is by calculat- ing the extent of eclipse according to true speeds and sara as explained later. Verse 6 : This book has used different methods for lambana correction for sphuta amanta (new moon day), true positions of sun and moon, dimensions of sun, moon and shadow, grasa (covered) amount of moon, sthiti (total eclipse time) 496 Siddhanta Darpana vimarda (total time of complete or maximum eclipse), real true lambana, sphuta nati, digvalana and parilekha etc. This will be useful, so learned men should not think it to be incorrect. Notes : Many of the methods have not been approved by earlier siddhanta works, but these methods give more correct results. Hence this needs to be accepted more eagerly, instead of rejecting it. His methods for different methods of moon's motion has already been mentioned in chapter 6. Correction of moon's and sun's motion is also due to his revised values of manda paridhis which change continuously in quadrants. For moon, only one maximum value has been indicated and its ratio with least value should be increased. Earlier, either the manda paridhi was fixed or a fixed difference of 40' was kept at the end of odd and even quadrants. Lambana and nati formula have been cor- rected due to changed formula of maximum nati. For moon this is taken as l/14th of daily motion instead of general formula of l/15th of daily motion for all planets. For sun it is entirely changed to 1/164 of daily motion, which has no parallel in earlier texts. The correct variation of nati and lambana has been calculated instead of rough linear method. Value of sun's diameter and consequently its distance from earth has been increased about 11 times the traditional value of 6,500 yojanas to 72,000 yojanas as mentioned in Atharvaveda. This has led to other changes in constants and methods. These corrections have been in right direction and more accurate. Lunar Eclipse 497 Verses 7-8 - Correct time of parvanta - On amanta or purnanta day (moon-sun = 0° or 180°), sun and moon will be made sphuta (at sunrise or midnight time. For parva ending, only mandaphala correction is needed in moon. On amavasya day, difference of moon and sun is taken, on purnima, it is moon - (Sin + 180°). Difference rasi etc is converted to para (1/60 vikala) and is divided by difference of sphuta gati of moon and sun in kala. Result will be in vighati (pala). This time in pala etc is added to parvanta time i.e. to sunrise time for which calculations had been done, if moon is less than sun (or sun+180° on purnima). If moon is more, it will be sub- stracted. Then we get the correct time (after or before sunrise for ending time of parva (purnima or amavasya). For this time, we again calculate sphuta moon and sun and from these values, correct parvanta time is calculated. After repeated applications of the method we get correct parvanta (for centre of earth). After that, other corrections for eclipse are made (like lambana or nati) for observation from surface of earth. Notes : As first approximation speed at parvanta is assumed to be same as at sunrise time and accordingly correct time is calculated. Our aim is to find the time when moon-sun or moon- (sun+180°) is zero. If moon is less than this value, it will cover up the distance due to higher speed. The difference is in para (1/60 vikala), speed diff. is in kala/day. Hence result time = _ f — kala/day 498 Siddhanta Darpana Kala x 60 X 60 _ X ^ a y - p a J a e fc kala J r After finding approximate parvanta time, we get better approximation of sun and moon position (their difference and their speeds. Then we get more correct value of parvanta. Vrses 9-11 - Samaparva Kala - When for sun, the mandaphala, gati phala and udayantara phala - all three are positive or negative, we further correct the samaparva kala i.e. middle point of eclipse is slighdy different from true parvanta above. Steps are as follows - (1) (Udayantara + bhujantara of moon) + (gati phala of sun) = S (2) S X mandaphala of moon = P (3) On purnima, X P moon diameter ( 444 yojana ) P On amavasya, X = Sun ^^^ (72/000 yo jana) X in vikala _ . , , . . (4) : : = L in danda pala etc. ' Moon gati - Sun gati (5) When mandaphala, gati phala and udayan- tara phala all are positive, Sama Parvakala = Parvakala - L When the three above are negative Samaparvakala = Parvakala + L (6) For this difference of time we further correct the positions of sun and moon at parvants. lunar Eclipse 4 " Figure 10 Notes : (1) Before analysing the formula we should analyse the reasons as to why closest contact will not be at amanta or purnimanta time. E is shadow of earth centre moving on ecliptic for lunar eclipse. For solar eclipse it is disc of sun. M is centre of moon moving on its orbit in direction MPN. At point EM, when EM is perp. to NE, ecliptic, longitudes of E and M are same which is ending time of amavasya or purnima as calculated earlier. However, closest approach is at P when EP is perpendicular at P. Thus the real mid point of eclipse will be after purnimanta time. When RM is after crossing N, then it is before parvanta time. This difference is due to inclination of moon's orbit with ecliptic and difference PM is given by udayantara phala of moon in latitude along ME direction and bhujantara phala in EN direction. Another reason of difference is due to different speeds at points of contact before P and after P. Due to that the mid point will be shifted from P in ratio of speed difference given by mandaphala of moon. Udayantara and bhujantara phala of moon are almost for same time difference as sun, as moon 500 Siddhdnta Darpana and sun or earth's shadow are in same position almost. The result for shadow at 180° from sun is same. If speed of moon is increasing, the time in covering contact distance towards N after P will be less and mid point will be towards opposite direction i.e. deducted. Similarly for other results positive, the time is to be deducted. If mandaphala is + ve, gati phala of sun is negative, hence relative motion of moon will be positive and it is to be added. Thus the difference due to latitude difference is (udayantara + bhujantara) of moon + gatiphala of sun. This will be increased in the ratio of mandaphala of moon. For outer contact, moon will cross its (own diameter + shadow portion). For inner contact (maximum) it will cover (shadow - its own diameter). Hence the product is to be divided by angular diameter of moon. In solar eclipse, it is almost equal to diameter of sun. There appears some error in text. All the quantities are in angular measure, which cannot be divided by yojanas, it should be angular diameter. When all the three factors causing error are of one sign, correction is proposed, otherwise they almost cancel each other. Qualiative discussion will be done at the time of calculating duration of eclipse. Verses 12-15 : Diameters and distances of sun and moon- In Atharvaveda, while explaining the meaning of 'Aum', diameter of solar disc has been stated to be 72,000 yojanas. Based on this statement, I Lunar Eclipse 501 have corrected the disc sizes of planets, their orbits etc. through observation and calculation. Diameter of moon and earth are 1/162 parts and 1/45 parts of sun's diameter. Earlier astronomers also have stated the diameter of earth as 1600 yojana (value obtained here). The values in yojana and angle are stated as follows - Diameter in yojana Angular diameter mean Sun 72,000 32/32/6 kala Moon 444 31/20 kala Earth 1600 — 72,000 yojana Mean sun diameter = Mean moon diameter = 2213 444 X 6 85 Mean distance of sun from earth = 76,08,294 yojana moon = 48,705 yojana. From this true distance, manda spasta karna also can be calculated. As in case of moon's angular diameter, earth's shadow's angular diameter also can be known in moon's orbit (approx by multiplying with 6/85). Notes (1) Comparative sizes of planets Aryabhata I, Lalla, Bhaskara I Sun's diameter Sun's distance 4410 Surya Modern values siddhanta, in yojana Siddhanta = 5 miles siromani • 6500 (6522) 1,73,156 6,89,378 1,85,80,000 502 Siddhanta Darpana Moon's diameter 315 480 430 Moon's distance 34,377 51,566 47,500 Earth diameter 1050 1600 1586 Diameter of earth is a measure of yojana as its astronomical definition. Hence; it is seen that diameter and distance of moon are almost accurate in surya siddhanta or others, but sun's diameter is taken only 4 times the earth or 14 times moon by Aryabhata (13.37 times by Bhaskara II or surya siddhanta). Its real value is 109.18 times earth's diameter or 402 times moon's diameter. However angular diameters were almost cor- rect. Bhskara II Surya Siddhanta Modern Siddhanta Darpana Values Moon 32/1 32/0 31/20 31/7 Sun 32/31/33 32/20 32/32/6 32/4 Angular diameters and their ratios are almost correct. Moon's angular diameter can be direcdy observed, but it is difficult to see sun directly. Still it can be seen through reflection etc, and due to frequent annular eclipses its mean diameter has been taken slightly more than moon. Linear diameter is calculated by formula (angular diameter x distance), when angle is in radians. This rato is almost 1/108 , this 108 is an important number for no. of beads in a prayer garland, no. of salutes to guru, astottari system of dasa in astrology etc. Moon's distance could be correctly estimated with direct parallax, but direct measurement of sun's distance cannot be done. The accurate looking figures of distances of sun and moon are derived from round figures of Lunar Eclipse 503 their circumference of their orbits after division by 2jt = 2 X 355/113 almost. On moon's orbit V has been assumeed eequal to 15 yojana by Surya siddhasnta and 10 yojanas by Aryabhata. Linear velocity of planet = (angular velocity X distance) has been assumed constant. Actually areal velocity = angular velocity X (distance) 2 is constant accord- ing to Kepler's laws for elliptical orbits. Thus all planets are assumed to cover equal distance in equal time and total distance covered by them in a kalpa is equal to orbit or circumference of sky. Accordingly, orbit of stars has been assumed 60 times orbit of sun. Candrasekhara must have seen distances of farther planets like pluto 40. times sun's orbit. Hence he increased it to another round figure 360 and explained difference of 1° sighra paridhi difference according to this, which is not correct. Similarly, he must have come across much larger figure of sun's distance and verified it according to parallax in solar eclipse. But he could increase it only 150 times moon's distnace instead of 400 times as he got diameters of 72,000 yojana from Atharva-veda. Earlier astronomers also must have obsereved it, but they didn't try to change it drastically, as the angular measure is sufficient for prediction of eclipse. Traditional value of surya siddhanta appears to be obstruction. Siddhanta darpana has assumed value of yojana in Atharvaveda as his own yojana which is incorrect as Aryabhata etc. had assumed yojana of about 8 miles; compared to 5 miles yojana of surya siddhanta. Of course, he has compared 1600 yojana 504 Siddhdnta Darpana diameter with surya siddhanta, though no such measure has been found in vedas. However M.B. Panta (Vedavati, Pune, 1981) has opined that for steller measures; maha yojana = 5 X Aryabhata yojana = 40 miles was used. Accordingly, Trisanku means 3 X 10 13 ; in mahayojana units it is 3X10 13 X40 miles = 207 light years which is really the distance of Trisanku star (Beta Cruris). Similarly Agastya or Argo navis has crossed Jaladhi or 10* 4 distance; which is 10 14 X40 miles = 690 light years in maha yojana units (correct distance is 652 light years). Mandala means revolution or circumference, diameter is indicated by width or viskambha in jyotisa. Hence 72000 yojana mandala means it is circumference. In mahayojana units this value means diameter of 9.1 lakh miles which is slightly more than 8.66 lakh miles, the modern value. This may be correct if we include the corona of sun. Another indication of yojana measure is given in Rkveda (1-123-8) Sayana has interpreted it that dawn goes ahead of sun by 30 yojanas and along with it moves round. Similar verse is in RK 6-59-6 which, dawn goes ahead 30 steps i.e. units of length. In modern astronomy, dawn is taken 18° ahead, Tilaka in his Arctic home in vedes, page 85, has taken it 16°, probably for central India at 24 °N. However, in sandhya of each yuga, its value has been taken as l/12th of yuga value. Thus dawn of day time of 12 hours is 1 hour, i.e. 1/24 of a day. This is 15° (360724) in angles. Thus circumference of earth is Lunar Eclipse 505 30X24=720 yojanas and sun's circumfereence is 72,000 yogana i.e. 100 times in round figures. In round numbers 108 japa is counted as 100 hence it gives almost correct dimensions of sun. Ratio of moon's diameter with earth's diameter has been slightly* increased and it is more correct according to modern values. Increase of parallax from 1/15 of earth radius to l/14th is also more correct and might have been confirmed by obser- vation. (2) Diameter of earth's shadow in moon's orbit - 85 yojanas in moon's orbit have been taken as 6 kala i.e. lkala = 14.2 yojana. Hence linear diameter of earth's shadow multiplied by 6/85 gives its angular diameter; because it is in moon's orbit. Verses 16-21 - True values of diameter and distance — If manda kendra (of sun and moon) is in 6 rasis beginning with makara, manda kotiphala is added to trijya and substracted from trijya if manda kendra is in other six rasis (karka to dhanu). Result is substracted from double of trijya, by remainder we divide the square of trijya (118, 844). Result will be sphuta manda karna of sun and moon. If this method is used for star planets like mangala, it will give their radial distance from sun as centre. This sphuta karna in kala is multiplied by madhya yojana karna and divided by trijya to give sphuta manda karna in yojanas. Madhya bimba kala divided by sphuta yojana and multiplied by madhya yojana gives sphuta bimba kala. (Quoted from Siddhanta Siromani) - Manda karna is found like sighra karna. It is substracted from 2 X trijya and by remainder, we divide square of trijya. Result in kala is manda karna of sun and moon which is the distance from centre of earth. 5Q6 Siddhdnta Darpana Manda karna kala multiplied by madhya yojana karna and divided by trijya gives sphuta yojana karna. Diameter of sun is 6522 yojana and of moon is 480 yojana (values of Bhaskara, not of this book - Quotation ends). Method of Bhaskaracarya also gives accurate value, still I have calculated sphuta karna from koti phala (instead of mandaphala because, for 3 raii difference between sphuta graha and mandocca, manda sphuta karna is equal to koti. Note : (1) True method - Madhya graha M is at angle from direction of ucca U. True planet S on manda paridhi has moved by same angle = ZSMN in opposite direction. SN is X on CM ex- tended. Figure 10 a NS = manda bhuja phala = r sin 6 where MS = r = radius of mandaparidhi R = 3438' = OM is radius of madhya graha. MS' // SN is mandaphala MS' _ OM R , NS " ON " R + r cos 6 because MN = r cos ON is called koti of karna, at 90° it is zero. Manda Karna OS = K is true distance of planet K 2 = ON 2 +SN 2 = (R + r cos<9) 2 + (r sin<9) 2 = R 2 + r 2 + 2 R r cos 6 (1) Lunar Eclipse 507 (2) Bhaskara approximation - His formula R 2 K " 2R - K appears meaning less as it can be used only if K is already known in right side also. However the first K is an approximation by koti of karna only = R + r cos 0. This relation holds good and gives a better approximation from formula. K 2 + R 2 = r 2 + R 2 + 2 Rr cos 0+ R 2 - - - from (1) = r 2 + 2R (R+r cos0) = 2 RK approx neglecting r 2 (3) Siddhanta Darpana formula has two unncessary steps for manda kendra 270° to 90 °, we first add manda koti phala to trijya, then substract the sum from 2 X trijya. This is equivalent to substracting mandakotiphala from trijya 2R - (R + r cos 0) = R - r cos 6 R 2 r> ,. r cos ^-i Now, = R (1 - — ^— ) l R - r cos 6 K r 2 cos 2 = R + r cos 9 + + (2) K 2r r 2 Now from (1), K = R (1 ■ + — cos + ^) V2 r 2 = R + r cos 6 + — + (3) average value of cos 2 6 = 1/2, hence, expression (2) is almost equal to K. Verse 22 : Mean angular diameters (bimba kala of moon and sun multiplied by true daily 508 Siddhanta Darpana motion and divided by mean daily motion gives true diameter in kala. Note : Linear diameter is fixed = D yojana Angular diameter B varies with distance, Bo is mean value Bo x R = BxK = D (1) True motion xK = Mean daily motion XR-(2) Dividing (1) by (2), we have Bp = B mean motion True motion „ Bo X True motion or B = (3) Proved mean motion v ' To prove (2), Let 0and & be the manda kendra for today and tomorrow at sunrise True longitude for sunrise today Rsinfl x R (Fig. 10a) R sin ' x R = Apogee today + arc , , I manda karna today True longitude tomorrow sun rise ( R sin ' : = Apogee tomorrow + arc manda karna tomorrow Taking difference of these two equations Daily motion for today = Daily motion of apogee (0 - 0) X R manda kendra for today a PP rox \ ) Here, manda kendra difference in one day has been ignored, (0'- 6) = daily motion of manda kendra i.e. mean daily motion. This is formula (2), if we ignore very slow motion of apogee. Lunar Eclipse 509 Verse 23 - Formula are Diameter i n yojana x R fa) Diameter in kala = — ~-j : w Spasta karna yojana This follows from (1) above. (b) B = Bo ± gatL £ Q f o r sun Addition is done when manda kendra is in 2nd and 3rd quadrant, otherwise substraction is made. _ B True motion Proof : or B Mean motion B - B True motion - mean motion B mean motion gati phala mean motion Bo x eati phala or B - Bo = 7 mean motion _ gati phala x 32/32/8 mean motion (59/8) (Putting values of Bo and mean motion) gati phala x 11 20 approx if both are in kala If gatiphala is in vikala then, the correction gati phala x 11 _ gati phala 20 x 60 " 110 a PP rox * Verse 24 : For moon moon eati phala , . b = bo ± ^^ (c) is 510 Siddhanta Darpana Proof - As in above formula correction is b x gati phala gatiphala x 31/20 mean motion 790/35 = gatiphala/25 approx ,^ „ True sun speed x 11 (QB = J? ^ v , True moon speed - 7 Proof (i) Formula (c) is obvious Bo x True speed 32/32 B = * = True speed x Mean speed r 59/8 = True speed X £ approx. _ bo x True speed _ True speed x 31/20 W ~ Mean speed ~ 790/35 True speed x 31/20 True speed - 7 = 31/20 x 25 -f 7 = 2^ a PP rOX as 7 is very small compared to speed of about 800 kala per day. Verse 25 : Shadow length of earth (conical from centre) True sun karna x diameter of earth Sun diameter - Earth diameter v Figure - 11 Lunar Eclipse 511 Note : In figure 11, BE is Parallel to AS, so AS SV SE + EV BE : SE + 1 or EV = SE x BE EV EV EV AS - BE Verse 26 : According to Siddhanta Siromani Diameter of earth's shadow in moon's orbit = Earth diameter — (Sun diameter -Earth diam) X moon distance sun distance Note : This is called reduction in earth's diameter; because sun is bigger and earth's shadow converges into a cone. Figure - 12 In fig 12, S,E, M are centres of sun, earth and shadow, Common tangent line ABC meets SEM at V. Radius of sun, earth, shadow are R, a, e. Distance of sun and moon from earth are r, n Shadow cone from moon MV = x. Now in similar triangles ASV and BEV AS BE SV EV or AS BE SE + EM + MV EM + MV R r + ri 4- x or — = a n + x n + x + 1 512 Siddhanta Darpana ri + x a ar or = or x = n - (1) r R-A R-a w In similar triangles BEV and CMV BE EV EM + MV CM " MV MV a ri + x r a or - = = — + 1 e x x ri a " e eri ,~ or — = or x = - - -(2) x e a - e Equating values of x from (1) and (2) ar e ri - ri = R-a a - e e _ ar _ ar - n (R - a) ° r a- e = n (R - a) 1 n (R - a) a ^ n (R - a) ar or — = 1 + e ar - n (R - a) ar__ n (R — a) ar-n(R- a) R-a or e = = a x n — (3) r r After multiplying by 2, result is proved. Verse -27 Earth shadow in kala Moon true motion 7 Sun true motion x 78 145 Note : Formula (3) in previous verse can be written as 2e 2a 2R - 2a ri (1) Lunar Eclipse 513 This gives angular diameeter in radians. Multiplied by Trijya = 3438' it will give diameters in kala. First term in kala in right side of (1) is 2a x 3438 ri ro True speed But — = * ri mean speed of moon (See equation (4) after verse 22), r = mean distance Hence this becomes 2a x 3438 X True speed r x mean speed 1600 x 3438 = " 48,705 X 790/35 X 1me Speed (Because 2a = 1600 yojana ri = 48,705 yojana, mean speed of moon = 790/35 kala True speed = -j — — approx. Second term in (1) is similarly (2R - 2a) x 3438 x True speed of sun Mean distance x mean speed of sun = True speed v (72,000 - 1 600) x 3438 7608, 294 X 59/8 &** ValueS 78 = 145 a PP rox * Hence the formula 514 Siddhanta Darpana Veieses 28-30 : Meaning of rahu Lunar eclipse is caused when moon enters earth's shadow, and solar eclipse is caused by covering of sun by moon. This is possible only when sun, moon and shadow of earth are near node (pata) of candra whose names have been given rahu and ketu (half part of rahu itself). Hence it is said that rahu devours sun or moon in eclipse. In siddhanta siromani - If eclipse is caused by same rahu, why there are different direction (of beginning of) eclipse, different times (short or long periods) and different coverings (total or partial eclipse). So persons assuming eclipse by rahu have false pride of their knowledge of sphere/ actually they are fools and against (true meaning of) Veda, purana and samhita. Rahu is shadow planet (a fictitous point), which covers moon by entering earth shadow (being near it); and covers sun by entering moon. For such ability, sun has given boon to him. This type of interpratation is not against scriptures. Veerse 31-32 : Reasons of eclipse Lunar eclipse - Shadow of earth is in opposite direction of sun and moves east wards in ecliptic like sun. At the end of full moon, when moon is in opposite direction of sun, its speed is more than shadow speed, so it enters the shadow and crosses it. After entring shadow, its light (from sun) is lost. Thus lunar eclipse is seen. Solar eclipse : At the end of amavasya, when moon and sun are in same direction (same rasi), then moon moving east covers sun and with faster speed crosses out in east direction. Sun being very Lunar Eclipse 515 luminous cannot bee seen. While covering, it, only lightless moon is seen. Verse 33 : Digamsa correction of rahu As we add or substract fourth phala (slghra correction) in pata of star planets like mangala, similarly digamsa phala of rahu is calculated as of moon and it is added or substracted. Note : Digamsa phala = 1/10 of mandaphala of sun. This is correction in moon's orbit due to variation in annual attraction of sun, which changes the direction of moon's orbit. Hence it changes direction of rahu also. This correction has been described in chapter 6. Verses 34-35 - Sara of moon Sphuta pala is deducted from sphuta moon' at corrected parvanta time. Bhuja jya of this arc is calculated. This is jya of viksepa kendra. We add 1/38 part of it. From half of result, arc is found. This arc divided by 6 gives sara of moon. When moon - pata (or vipata candra) is in six rasis beginning with mesa, sara is in north direction, otherwise in south direction. N N' m S Q Q' Figure - 13 Notes - Maximum sara of moon has been stated as 309' in siddhanta darpana i.e. inclination angle e = ^MNS is given by R Sin e = 309' NS 526 Siddhanta Darpana is ecliptic on which S is position of Sun or earth's shadow. NS = m = distance of moon from node N. NM is orbit of moon with moon at M. Its sara is MS = p. p = m tan e as AMSN is right angled and almost plane due to small size. , m . R sin e m . R sin 309 US P R cos e R sin (5400 - 309) m x 308 m ' „ x = ~^iii — = — r approx - ' ■ (1) 11 + - 3 However, at the time of eclipse, S has slow motion and is considered fixed and we calculate only the moon's speed. Relative speed of moon is obtained by adding vector VV 1 equal and opposite to motion of S to velocity vector MV of moon. Thus resultant motion of moon is smaller and in direction MV'N' which make angle e' slightly bigger thane' VQ motion of sara tan e = tan e' = QM motion along ecliptic VQ or QM - VV tane' QM tane QM - W 790/38 X cos (309) or tan e' = tan (309') „ nn ,^ n ,„ rtrtX v ' 790/38 cos (309) - 59/8 thus e' = 333 = 5° 33' Shortest distance of moon from ecliptic = p cos e' = SP which is pependicular from S to MV' Lunar Eclipse si7 Hence effective sara = p cos e' _ p . R . sin (5400 ^ 333) P x 3420.5 R " 3438 (2) Equation (1) takes value (3438/3423) times more than the sine value. For half the angle increase is about 1/38 times as approximated here. Hence after increase of 1/38 in m, Sin,, of its half value in taken, then divided by six again. Taking sine almost equal to small angles, formula given is 39 1 13 P = ^8 X U m = m m which is almost same as (2). as may be verified. Due to relation (2), the effective inclination of moon's orbit is reduced by about 18' to 290' approx. Hence the value of parama sara was taken as less than the true value, in earlier texts. Verse 36: Method for sara gati Instead of finding arc, we multiply the kotiphala of moon and it is multiplied by pata and motion of moon and divided by trijya. Result is current speed of sara. By adding or substracting kranti gati, we get sphuta sara from equator. Notes: (1) There are three confusing words in the verse-Whose kotiphala is to be taken is not specified — I have interprated it to be kotiphala of moon's movement along ecliptic i.e. its rasi etc from pata. Whether motion of pata and moon both are meant — pata has very small motion and when motion sun is being neglected, much smaller motion of 518 Siddhanta Darpana pata cannot be taken. Hence it is pata or sara from ecliptic and motion of moon. Result of this multiplication and correction with kranti both are called sphuta sara. First sphuta sara is distance of moon from kranti vrtta. Second sphuta sara is distance from equator = distance frojm kranti vrtta(sara)+ distance of spasta moon on kranti vrtta from equator (i.e. kranti of moon). Translation has been made according to these clarifications. (2) Sphuta sara from equator has already been explained. Now p=sin m. tan e. p=sara, m= distance of moon from pata along ecliptic £ = angle of inclination of moon's orbit with ecliptic, since e is constant, taking differentials Cos p. (3p= Cosm. <5m. tan e Here dp and (5m are motion of pata and moon in unit time of hour or day. We are to find dp. Rcosm K tan £ dp = ~ .(5m. — — (1) r R cos p Now cos p= cos e/sin m according to Napier relations of right angled triangle N E P. tan £ Hence, = sm m sin e ~ sin p ~ p cos p Hence (1) becomes motion of pata Kotiphala of moon x Moon gari ,x Pata Tnjya verses 37-38: sara from chart Pata is substracted from moon and bhuja of the resulting angle is found. From degrees of bhuja Lunar Eclipse 519 sara etc can be found in charts where sara, sara difference, kotiphala and bhujaphala etc have been given in appendix: These have been given at intervals of 225. Alternatively, pata is substracted from moon. Its bhuja is converted to kala. Its 1/16 will be subtracted and divided by 11. (i.e. moon-pata). For greater values, this will be incorrect Notes : Kala of bhuja is almost equal to jya for small angles. It is to be multiplied by 309 Parama Kranti 3438 1,c " 1 Trijya 1 -., 39 309 1 11 + 8 approx. Substraction of 1/16 part is to convert the bhuja approximately to its jya: Verse 39: Extent of eclipse (grasa) The planet which is to be eclipsed is called grahya or chadya and the planet or shadow which covers it, is called grahaka or chadaka. Half the sum of their angular diameters is 'manardha' or 'manaikyardha' . If sara is more, there cannot be eclipse. Differance of manardha and sara is the grasa (covering). If grasa is more than grahya, then eclipse is total. Remaining part of grasa is in sky. Note : Except for the terms, the cause of eclipse has already been explained. 520 Siddhdnta Darpana M Figure 14 (a) Figure 14 (b) P is centre of planet to be eclipsed and XM is diameter. Its distance from centre of coverer O is sara-OP. Either the coverer (chadaka) earth shadow or covered (chadya) sun is on ecliptic In figure 14(a) covered portion XY = PX - PY = ri - (OP - r 2 ) = ri + r 2 - OP Where rl and r2 are radii of the bimba. When XY > 2ri then eclipse is total as in fig 14(b) Verse 40 : Direction and stages- Lunar eclipse has contact in east and end in west often (as explained in verses 31-32) Calculation is done as per this rule only. But solar eclipse often starts in west and ends in east direction, However, sometimes south west direction is calcu- lated instead of east west direction When grahaka just completely covers the grahya, it is called 'nimilana' time. When it is about to start emerging, it is called unmilana. Time from nimilana to unmilana is called 'marda kala' or 'vimarda kala' (period of total or maximum eclipse). Total time of eclipse from sparsa to moksa is called grahana kala or 'sthityardha' kala. lunar Eclipse 521 verses 41-43 : Total time and time of complete eclipse- Parvanta time. When moon and sun have same rasi etc. (in solar eclipse) or their difference is exactly six rasis; it is the time of eclipse. This time (after minor correction of verse 11) is called sama parva kala and is middle point of eclipse. Square of manardha (half sum of diameters of grahya and grahaka) is substracted from square of sphuta sara and of remainder, square root is taken. This is multiplied by 60 and divided by difference of moon gati and sun gati. Result is half time of eclipse (sthityardha kala). Its double is total time of eclipse. Similarly, square of half the difference of diameters of grahya and grahaka is substracted from square of sphuta sara. Square root of remainder is multiplied by 60 and divided by differance of gati of moon and sun. Result is 'marda- ardha' kala in ghati etc. Its double is 'marda' kala or time of complete eclipse. From samaparva kala (mid point of eclipse), substraction of sthiti ardha and marda-ardha give sparsa and nimilana times. When sthiti ardha and marda-ardha are added, it gives unmllana and moksa times: For more correct times, we calculate sphuta candra and sara at time of sparsa, unmllan etc and from them again these times are calculated. In solar eclipse, repeated larhbana corrections are made. Notes : (1) This is an approximate formula in which sara of moon is considered to be same, hence there is need for successive approximation. First, we derive the approximate formula, assuming 522 Siddhdnta Darpana the sara to be minimum distance of moon from mid point of eclipse. Figure 15 - Period of eclipse In figure 15, O is centre of earth shadow and ON is ecliptic. Shadow is considered fixed and moon is moving in direction Mi N with relative speed (moon-sun). This direction is slightly more inclined 5° 33' compared to 5° 9' angle of moon's orbit with ecliptic, as explained in verse 35. Ml, M2, M3, M4 are positions of moon at 1st contact (sparsa), 2nd (nimllana), 3rd (unmilana) and 4th contact (moksa) OP is perpendicular on Mi N and is almost equal to sara. This value of sara is assumed for all the four positon. Right angled triangle OPMi is almost a plane figure. Hence M i p = VMi O 2 - OP 2 = V l(moon bimba + shadow bimba) z - sara 2 2 Moon moves along MiP with relative speed of m-s where m and s are daily motions of moon and sun(or shadow) in 60 danda. u •. mi „„. 60xMiP Hence it will cover MiP in danda m — s This is same as M4P distance (as Mi 0=Mi O). 523 lunar Eclipse Similarly, OM 2 or OM 3 = radius of shadow- radius of moon. P M 2 or PM 3 = VoNg - OP 2 I "— m = V 1 (shadow diam - moon diam.)* - sara' 2 Hence half time of total eclipse 60 m — s — -~ ^manantara 2 - sara 2 — C (2) Let T be time of conjunction, when moon and earth shadow have same longitude, and p the latitude (sara) of moon, North latitude is considered positive, p' is hourly increase in latitude (increase towards north is positive) m'= excess of hourly increase in longitude of moon over that of sun. M= angular radius of Moon, S= angular radius of shadow at the moon. Then at any time t hours after time of conjunction, T, the distance between shadow and moon in longitude is m't and the latitude of moon is p+p't. Thus distance between centres of shadow and moon ={m ,2 t* + (p + P' t) 2 l 1/2 The eclipse begins or ends when the moon's rim just touches the rim of the shadow in entering it or leaving it. Distance between such time is S+M=D say (fig 15) then { m' 2 t 2 + (p + p' t) 2 } m = D gives the time of beginning of eclipse. Solving this for t, we get 524 Siddhanta Darpana t = pp m /2 + p' 2 « 2 w 2 P P ./2 ,ia + (m" + p' z ) D 2 -p 2 m' 2 + p' 2 iv 2 The + sign gives the end and -sign (earlier time) gives the beginning. Total phase of the eclipse begins or ends when the rims touch the moon being inside shadow (M 2 M3 position of fig. 15) i.e. D= S- M, Putting this value of D in above soulution, we get the times of beginning or end of total phase of eclipse. Discussion of results: (1) The eclipse begins at T- PP m' 2 + p /2 P 2 P' 2 ,2 ao. + (m" + p") D 2 -p 2 m' 2 + p' 2 Vi (2) Eclipse ends at PP T- m' 2 + p' 2 + p 2 p' 2 rr + D^-p' Vi (m' 2 + p' 2 ) 2 m' 2 + p' 2 j hours For full eclipse time D=S+M. For total eclipse D=S-M. r (3) Middle of the eclipse falls at PP' T '2 7^ hours m' z + p' 2 (a) If p and p' are both positive or both negative, middle of the eclipse is before the time of conjunction. (If one is positive and other negative, middle is after conjunction) (b) Only when the latitude at conjunction p=o, the middle falls at T, the time of conjunction, because p' cannot be zero near a node. Lunar Eclipse 525 In verse 11— p latitude is positive then udayantara is positive, p' i.e. speed of latitude is positive when bhujantara is positive (verse 36). Thus when both are positive or negative time is corrected (4) If D=p, (1) and (2) reduce to t PP' -+. PP' 1 — 1 n — m' + p' m' + p' (a) Eclipse begins or ends at the conjunction (b) Duration of eclipse is — 2 2 which may m' + p' amount to about 22 minutes (5) The duration is zero, when the expression between the double brackets is zero, i.e. p is greater D' 2 than D by , 2 &r , (neglecting fourth powers of p'/m'), which may amount to about 14" in the mean. (6) If t is not real, there is no eclipse, or total eclipse, according as D is taken to be S+M or S-M. Then p-D is more than about 14" calculated above (3) Conditions of eclipse in equator coordinates Instant at middle of eclipse is chosen as origin of time in figure 16. Equator Figure 16 526 Siddhanta Darpana Equatorial coordinates of centre C of shadow at time t hours is a, d and of centre M of moon be ai, di. Then, if P is the pole and M D the perpendicular from M on P C (on celestial sphere), CD= <5i-<3 and DM= (ai-a) cos <5i, nearly. So CM 2 = (di - a) 2 + (ai - a) 2 cos 2 <5i - - (1) If, hourly rates of increase of a, ai, 6, d\ at t =o are (a) o, (ai) (6) (^i)o respectively, we can write (1) as CM 2 - [{ (dOo + d\ t } - [(6)0 + <5' t}] 2 + [ {(al) + a' t } - { (a) + (a' t) } ] 2 cos 2 <3, (2) approximately, neglecting the changes in cos (3i due to changes in 6\, because cos 2 6\ in the above equation is multiplied by a factor which is small. Equation (2) is of the form CM 2 =a t*+ bt +c (3) Where a, b, c are known quantities. 51 If we put CM= — (P + Pi - S) + S 2 the two values ti and ti given by (3) are the times of 1st and fourth contacts (sparsa and moksa). For the second and third contacts (i.e. beginning and end of totality, we put 51 CM = — (P + Pi - S) - Si and solve for t. w/V/ 1 Middle of the eclipse is — (ti + ti) 2 K± *-' 2a verses 44-45 —Single time calculation Method above uses successive approximation. Now method of single time calculation is described. Sara of samaparva kala is calculated. Its half is Lunar Eclipse 527 divided by difference of gati of moon and sun; Result in danda etc is substracted from samaparva kala sara, if sara is increasing, otherwise it is added. For this parvakala, new values of moon and its sara are found. Diffenence of this sara and samaparva kala sara in vilipta is squared and its half is taken. Its square root substracted from samaparva kala sara is sphuta sara. Sthiti kala calculated from this is correct. Now more accurate value of shadow is stated (in verses 78-84) Notes — In figure 16(a). AB is ecliptic and CD is moon's orbit, relative to shadow centred at S on the ecliptic. S and M B A N2 N1 S Figure 16a - One time calculation of sthiti ardha are the centres of shadow and moon respectively at the time of oppositon. SMi is perpendicular from S on moon's orbit and MiNi is perp. from Mi on the ecliptic. Then Mi is the moon's centre at the middle of the eclipse. AMMiS is almost plane, ^MMiS=90 MS = moon's latitude at opposition, Z.MSM1 = i, inclination of moon's orbit to ecliptic. Ni S = Mi M approximately as i is small = 3 ° 9 * MS minutes (Kala) as R sin i = 309' 3438 v ' 528 Siddhanta Darpana 309 x 60 x MS 1 , j _ TTTr x : — ; : danda 3438 gati antara (of moon and sun) MS . — : approximately (1) 2 x gati antara vr J _ MS x 309 x 60 x 60 or Ni S - 343g (790 , 35 „ _ 59 , 8 ,^ pa a MS i n\ = — palas (2) This time is subtracted from the sthitiardha. Since square of sara is used in calculation, average of squares of sara at M and Mi is taken. Hence, half the square of difference is taken. verses 46-50 : Grasa from time. Now method is described to calculate grasa from time and vice versa. If time is before mid eclipse, it is substracted from sparsa sthiti ardha time. Remainder in danda etc. is multiplied by hara =(moon gati-sun gati) corrected for lambana for solar eclipse, next chapter verse 46-47) and divided by 60. This will be koti kala of lunar eclipse. In solar eclipse, it is again multiplied by madhya sthiti ardha and divided by sphuta sthiti ardha, to get sphuta koti kala. For given time, squares of koti kala and bhuja kala are added, square root of sum is karna. This karna substracted from half the sum of bimba kalas gives grasa. If the given time is after mid eclipse time, it is deducted from moksa sthiti ardha. Difference is multiplied by gati antara of sun, and moon (hara) and divided by 60. We get koti. Then sara of given lunar Eclipse 529 time is found, from which spasta koti kala of solar eclipse can be found. Again karna is found by adding the squares of bhuja and kotikala and taking square root. Karna substr acted from half sum of bimba, gives grasa. From grasa value, remaining free portion of eclipsed planet can be found. Notes . (1) Grasa = covered part (literal, meaning devoured portion) Amount of grasa is the length of diameter along the line joining centres of covered and covering discs, which has been eclipsed. Magnitude of eclipse (in modern astronomy) is grasa expressed as fraction of diameter. Thus grasa = radius of shadow + radius of moon -distance between centres of shadow and moon Magnitude = grasa/diameter of moon. For solar eclipse, instead of shadow, we take moon's disc and covered disc is of sun. (2) Formula of grasa has already been estab- lished while calculating sthiti or marda times. To revise, refer to figure 15. If M is any position of moon's centre, MP is distance covered from central point P. If it is before P, then it is at Mi (contact point or sparsa). Then in the given time after sparsa, moon moves from Mi to M. The remaining portion is MP till mid time at P. Thus in time (sthiti ardha-given time) =t r planet covers MP which is — — — where m' is 60 difference of daily speeds of moon and sun. 530 Siddhanta Darpana m/60 is speed in one danda. MP is koti kala OM = Vqp 2 + MP 2 = V m't 2 ^ = Kama U1 ^ 1VU (~7Z) + (sara) 6U 1 When OM < — difference of diameters, com- plete portion of moon is covered. For OM bigger than this value l/2(sum of diameters)— OM is amount of grasa. Similar calculation is done for period after midtime. (3) In solar eclipse there is fast change in sara and valana, hence true kotikala is found. spasta koti kala _ madhya sthiti ardha Madhya koti kala spasta sthiti ardha Because, if sthiti ardha increases, the dif- ference with given time decreases and koti kala decreases. Thus, they are inversely proportional. Verses 51-53 : Time from grasa When grasa is between sparsa and mid time, then it is substracted from half sum of covered and covering discs. This gives difference karna between centres of two discs. From square of this karna, we substract square of spasta sara at that time. Square root of difference will be koti kala. For solar eclipse this kotikala is multiplied by lambana corrected sthiti ardha and divided by madhya sthiti ardha. This gives spasta koti kala. This is multiplied by 60 and divided by difference of daily speeds of moon and sun. Result in danda etc. is the time after sparsa. Lunar Eclipse 531 9 For position in second half of eclipse, the result is substracted from sthiti ardha time, The remainder will be time remaining till moksa. Notes : This is reverse process of the previous method and uses the same formula. Verses 54-Method for solar eclipse-For solar eclipse, the sthiti ardha for sparsa and moksa is called mean sthiti ardha, because special paral- lax(lambana) correction is done in this. Hence, all processes are done with mean sara (this doesn't change in short period of eclipse). Repeated parallax correction will give correct time. Verses 55-59 :Direction of eclipse from parallax— Now I describe valana(parallax) correction in kalas for correction of moon and sun in their eclipses, which arise due to ayana and aksamsa. Due to these effects, direction of sparsa, mid-point and moksa of an eclipse is known in east or west portions (kapala) of sky. In case of lunar eclipse, sayana candra, and in case of solar eclipse, sayana sun is found. Its kotijya (in kala) is multiplied by parama kranti (1410) and divided by 3 rasis (5400kala). Result is ayana valana. This is in same direction (east or west part of sky) in which eclipse takes place. In solar eclipse, rasi of sun and moon is same, hence ayana valana can be found only from moon. We calculate the nata kala in pala from moon midday in lunar eclipse and from solar midday in solar eclipse This multiplied by 90° and divided 532 Siddhdnta Darpana by its half day time gives nata in east or west direction in degrees This nata is multiplied by aksamsa of the place and divided by 90. Result will be aksa valana in north direction for east nata direction or valana in south direction for west nata. Aksa and ayana valana are added when in same direction and difference is taken for different directions. Result will be dik- valana in degrees of moon in lunar eclipse and of sun in solar eclipse. This is true valana from which sparsa and moksa directions can be known. Its measure in angula has been stated while describing parilekha (degrees). Notes (1) Sphuta valana is the angle between east or west point of disc of eclipsed planet with kranti vrtta (ecliptic). This is made of two com- ponents. Due to aksamsa of the place (distance from equator), kranti vrtta cuts the horizon in eastern half of sky in north direction from east point. So ecliptic is towards north of east point of disc in east half of sky (and towards south in west half). This is called aksa valana. Due to angle between ecliptic and equator (causing ayana), ecliptic is inclined further towards north when say ana makara is on meridian (north south vertical circle). When sayana makara is on meridian (±90°), it is shifted southwards in east half of sky. For west half, the directions are opposite. This component is called ayana valana. Lunar Eclipse 533 E' E K Figure 17 - Ayana and Aksa valana NVZS is yamyottara vrtta (meridian) at desired place. NES is east half of horizon (ksitija vrtta), north, east and south points shown Z=kha-swastika (Zenith), ZE = Samamandala, Z'E' parallel to ZE through C,- e, w — are east and west points. KK' = Kranti vrtta C= centre of planet disc to be eclipsed (chadya) NCS = Samaprota vrtta of c (circle of position) V = North Pole in sky P = pole of ecliptic (kadamba) P P1P2 is kadamba vrtta in which P moves round V in a day. Z C 1 =Nata degree (in time units) of C P2 = Kadamba when sayana karka is at K' (meridian) Pi = Kadamba when sayana makara is at K' CV = Polar distance, C P = Kadamba distance 534 Siddhanta Darpana ZNCV = = Aksa valana, ZVCP = 0'= Ayana valana = Ayana valana L NCP = 6 + 6' = sphuta valana = L KCE' In this figure 17, for position P betweeen ± 90 Q distance of Pi, ayana valana 6' is also in north direction, hence sphuta valana is 6 + & as shown in figure. For P between ±90° of P2, 6' will be in south direction and sphuta valana will be 6-6'. The direction of valana will be opposite, when planet is in west kapala (west half of sky). (a) Aksa valana - From spherical triangle NCV Sin NCV _ sin CNV sin ZC sin NV sin CV sin (polar distance) because NZ and NC both are right angles, hence angle between them is equal to ZC, which is natamsa of planet. sin (polar distance) = R cos 6, (d - kranti of planet) = Dyujya or radius of ahoratra vrtta. sin NV = R sin O , O = aksarhsa of the planet. so, R sin x sin ZC R sin x sin ZC sin NCV = ^— r-- = £ j (1) Dyujya R cos o Rule for finding natamsa— This is as per Bhaskaracarya. In half day or half night time, a planet rises 90° from horizon, hence nata kala multiplied by 90 and divided by half day (or night in lunar eclipse) time gives natamsa in degrees. This is not the angle from vertical Z point, but the angle between meridian lunar Eclipse 535 and samaprota vrtta, corresponding to nata kala. (H) Its relation with natamsa from Z is z = ZC. This can be found from spherical triangle NZC, cot Z C x sin ZN =cos ZN. Cos NZC + Cot ZNC. sin NZC But ZN = 90° hence sin ZN =1, cos ZN = Hence, cot ZC = cot ZNC. sin NZC ^^ cot ZC or, cot ZNC = — — rr=^ ' sin NZC or, tan ZNC = Sin NZC. tan ZC But, L NZC = 90° + L EZC = 90° + agra Hence sin NZC = cos (agra) Hence tan ZNC = cos (agra) x tan z Rule for aksa valana : Surya siddhanta and Bhaskara II both have given the formula (1) i.e. Jya of natakala is multiplied by Jya of aksamsa and divided by dyujya or semi diameter of diurnal circle. In this text, R sin NCV and R sin <p both have been approximated to the angles NCV and <p and dyujya is equated to 90°. When d is small, R cos 6 = R sin 90°. nearly. Thus all the 4 jya are slightly increased to the arcs and the errors almost cancel each other as a rough rule. (b) Ayana valana is known from spherical triangle PCV in figure 18 Sin L PCV ■_ .sin CPV sin PV " sin CV * sin PV x sin CPV or sin PCV = : — ^77 sin CV 536 Siddhanta Darpana Figure 18 PV= distance from dhruva to kadamba which is equal to parama kranti (angle between ecliptic and equator). CV is distance from dhruva whose jya is kotijya of kranti LQ P V is the angle between circles from C to ecliptic pole N2 and ay ana circle Ki P. Positions of planet on ecliptic and equator are Li and L 2 . L CPV=arc Ki Li = 90° - ML! where M is vernal equinox. Hence Jya of CPV is kotijya of MLi = sayana graha Jya of parama kranti x kotijya Kotijya of kranti This is the formula given Verse 60-65 : Period of lunar daytime: Solar day time has already been described. The period from moon rise to moon set is its day. At sunset time, sphuta sayana sun and moon are calculated. For sun, rising time (in asu) is calculated for remaining part of rasi and for moon, it is for lapsed part of rasi. These two udaya asu are added with rising times (udaya asu) of the rasis from sun to moon. We add 56 asu to the total and divide by 360 to make them ghati. This time after sun rise moon will rise. For finding moon-set time, sayana sun and moon for next sun rise time is calculated. Rising time of rasis between sun and (6rasi + moon) is Thus sin PCV lunar Eclipse 537 calculated and 56 asu lambana time is substracted. This time after sun rise, moon will set. Sara of moon is very little (within 5° 9' and almost zero at eclipse time). Hence time between rising and setting of sun will be its day time, which has been calculated for diurnal circle of sun. Sara in kala (minutes of angle) at rising time or setting time is multiplied by palabha and divided by 12. Result will be added to rising time if sara is south and substracted if sara is north. Reverse is done for correcting moon-set time. Thus we get sphuta time of moon rise and moon set. Alternatively, sphuta gati of moon at midnight is divided by 19 and result in pala is added to night time of sun. This gives day time of moon. Notes : (1) Difference between rising times of sun and moon is the difference between rising times of their rasis, since sun and moon move in almost same ahoratra vrtta. Sara at eclipse time is almost zero. Since, at purnima time, moon-sun is less than 180° (it is 180° at end of purnima), when sun has risen, moon will be slightly above west horizon. Thus difference of moon from sun +180° or (moon +180°-sun)distance is to be covered by moon for setting after sunrise. Due to parallax, angle of moon at horizon seen from surface is 56' lower than the angle calculated from earth's centre. Thus moon will rise on horizon after covering 56' more. Hence moon rise time will be later than the rising time of rnoon-sun by further rising time of 56'. 538 Siddhanta Darpana Similarly setting of moon will be earlier by corresponding rising time of 56' extra arc. (2) Alternative formula — Solar day in asu is more than naksatra day in asu (21, 600) by the daily motion of sun (59.8"), extra time taken by earth to cover this distance covered by sun in mean time. Similarly, lunar day is more than naksatra day by its daily motion in asu i.e. 790735" It is more than solar day by 790 (1-1/13.37) asu For true speed, it is more than solar day by moon gati (1-1/13.4) asu, relative speeds of sun and moon assumed almost content Due to parallax the decrease in day time is (moon gati/14) both at moon rise time and moon set time. Hence (moon day-sun night) moon gati ' 1 ^ ^ - l - l 2 / 13.4 2 x moon gati asu (Moon day = - moon - day and night ) = moon gati 12.4 \ = moon gati x 2 x 6 13.4 6.2 13.4 1 7 \ asu 7 V pala moon gati = V — Pala approx. Verses 66-69 : Explaination of valana correction. On great circle from north pole to south pole in the sky, pole of ecliptic called 'Kadamba' is situated 23° 30' south from north pole. This is surface centre of ecliptic in north part of celestial sphere. Lunar Eclipse 539 The south surface centre of ecliptic (kranti vrtta) is called 'kalamba' which is north from south pole by same 23° 30' angle on ayana prota vrtta (between two 'dhruva') 'Sara' is calculated along kadamba prota vrtta which is distance from ecliptic. Moon disc moves fastest of all the planets. Hence only its difference along two circles ayana prota and kadamba prota is calculated. Distance of moon from 'dhruva' on dhruva prota vrtta and from 'kadamba' along great circle through kadamba is taken. Their difference (an- gular) is multiplied by 360 and divided by circumference of moon disc (angular). This gives ayana valana. When moon is in north ayana, it is north valana and it is south valana in ' south hemisphere from equator. For aksa valana, Lalla and Srlpati have calculated versine of nata. But it has been done from R sine of nata by Brahmagupta and Bhaskara II. For ayana valana also two methods exist. One is from kotijya of madhya graha and the other from versine of bhuja of sayana graha. But in my method, no jya is needed because nati is according to equator and ecliptic arcs. Hence koti degree and nata degrees only should be used for ayana and aksa valana. Note : Correct method and meaning of terms has already been explained. Verses 70-77 : Diagram of eclipse— For making a parilekha (diagram), place is made plane like water level and a circle of 18 angula semi diameter is drawn. East and west points are marked as explained earlier (in Triprasna 540 Siddhanta Darpana dhikara) From these two points on circumference also two circles touching each other are drawn, each of 18 angula semi-diameter. In these two circles also, 4 points for cardinal directions and 4 middle angles are marked. An east west line is drawn through centre of the two circles. A point is marked 1 angula north of north point of eastern circle and another point 1 angula south of south point of western circle. When planet is in west kapala (west half of sky), a circle of 16/10 angula semi-diameter is drawn from southern point. When planet is in east kapala, same size circle is drawn from northern point. These arcs in the respective circles indicate kranti vrtta (ecliptic) Both arcs in the respective circles indicate kranti vrtta. In that signs of 12 rasis from mesa are given from west to east after making 12 equal parts. Centre of moon is kept in its correct rasi of kranti vrtta and around it, a partly eclipsed moon circle is formed. A line joining its two horns is drawn. The . line joining horns is equal to diameter of moon. With this diameter, circles are drawn at both external points of kranti vrtta. From this diagram moon will appear to be moving on kranti circle. In eastern circle, kranti vrtta is 328 yojana north (23 -° aksamsa of karka rekha) from equator which is line between east and west points of the circle. Kranti vrtta is actually a straight line, but appears curved due to drawing in a plane figure. Lunar Eclipse 541 Hence jya or kotijya are not needed in aksa or ayana valana. The curved shape kranti vrtta (and equator also) is perpendicular on all yamyottara (meridian) lines between two poles. Hence, on this kranti vrtta, distance from prime meridian (Ujjain or Greenwich) is desantara jya. Similarly, aksa jya is distance on north south line. Notes : This is like representation of earth in two touching circles in which karka rekha and makara rekha are north and south of equator. N Figure 19 In figure -19 central circle is only for finding east west direction. East, west circles are of 18 angula semi diameter in which all directions have been marked. P' is 1 angula north of N 1 , P 1 is 1 angula south of 5. Kranti vrtta. Qi Q2, Q3, Q4 are drawn from these with 16/10 angula radius. This is only for explaination and not to the scale. However, this is a copy of school atlas map and reasonings about aksa valana and ayana valana on tn at basis are not correct. 542 Siddhanta Darpana Verses 78-79 : Effective shadow of earth. In moon's orbit, there is 5 kala less dark shadow (avatamasa or penumbra). On adding this, earth's shadow diameter increases by 10 kala. This semi dark shadow covers moon at other times also, then there is no eclipse but light of moon is dimmed. 1/3 part of this semi shadow (penumbra) is very dark hence it almost merges with main shadow. Hence 1/3 of avatamasa or 10/3 kala is added to the earth's shadow to find the effective diameter of shadow. Notes: (1) M2 M3 penumbra in moon's orbit is formed F Figure - 20 by direct tangents GB and transverse targent FB' (this will be very close to B). Z.FBG = 2R/r, where r is distance of sun R = radius of sun. Hence penumbra at distance ri of moon, making same angle. M2 BM3 is 2Rri ri x L FBG = yojanas 6 2 R ri x Kala 85 Lunar Eclipse 543 6 72000 x 48,705 _ ,. ' , = — x ^7~^r~^r* = 32.5 kala 85 76,08,294 Thus the extent of lesser dark shadow is arbitrary. However, in penumbra, moon's light will be definitely lesser. As explained earlier, the effective increase of earth's shadow is by 2% or about 1 kala due to absorption by atmosphere. Verse 80-81 : Size of earth's shadow- It changes both due to sun distance and due to moon's distance, where its size is calculated. When sun is near mandocca, it is farthest from earth, hence shadow is bigger when gati is small and at 90° from nica, it reduces. Hence 1/28 of ravigati phala is added to shadow or substracted from middle value. Moon's diameter is multiplied by 35 and divided by 13. In this, gati phala is substracted when positive. This gives true value of earth shadow. Method to find moon's diameter has already been stated. Note : True dimensions of shadow has already been stated based on true motions of sun and moon both in verse 27. This correction is based on arbitrary assumption of avatamasa' i.e. darker part of penumbra. Verses 82-83 : Calculation of true earth shadow-Due to relative rotation of sun around earth, earth shadow also rotates in same directon ^th same speed, but always remains opposite. It covers moon according to its value in moon's orbit, here is difference of 1/20 parts due to variation m ^stance from sun. Due to varying distance of 544 Siddhanta Darpana ; moon also its value changes. But this is very small compared to variation due to sun, hence it is neglected. Now method to calculate effective earth shadow is explained. From sun's diameter (72,000 yojanas), its l/10th (7,200yojana) and earth diameter \ (1600yojana) are substracted. Remainder (63,200) is [ multiplied by mean moon distance (48,705 yojana) to get (3,07,81,56,000). This product is divided by I true distance of sun. The result substracted from earth diameter is the diameter of shadow in moon's orbit. This diameter multiplied by trijya (3438) and divided by true distance of moon gives angular diameter. Note : 'Avatamasa' (dark part of penumbra) • is 10/3 kala which is 1/12 of earth shadow (about 40') Hence (sun diameter-earth diameter) is reduced X by 1/10 of sun diameter. Rest of the process is already explained in verse 26, whose diagram will] make it clear. Verses 84-86 : Colour of eclipse From the shadow of earth, 40 kala deducted! gives the value of andhatamasa (dark penumbra). \ (shadow is as calculated above). When sara of moon j is small, moon enters this dark penumbra and| . -J looks very dark. When lunar eclipse is very little, sky turns blue. In half eclipse, sky appears black. In morej then half eclipse, it looks red black. In total eclipse, moon becomes pale yellow due to its entry in earth's shadow. In solar eclipse, there is no change in colours; we seen only moon which is relatively dark. Lunar Eclipse 545 Moon is always smaller than sun, even in angular diameter. Hence horns of sun are sharp in solar eclipse. But moon is cut by bigger circle of earth's shadow, so its horns are rounder in eclipse. Note : This is subjective description, hence no comments Verses 87-88 : Close Being dark in colour, shadow of earth is like rahu, in which moon enters at eclipse time and gives mantra siddhi to vaisnava and tantrikas. They may do good to us. Thus ends the eighth chapter describing lunar eclipse in detail in siddhanta darpana written for education of children and correspondance between theory and observation by Sri Candrasekhara born in famous royal family of Orissa. Eighth chapter ends. Chapter - 9 SOLAR ECLIPSE Solar Eclipse Verse 1 - In last chapter, eclipse of moon and sun both have been discussed in a general way. For solar eclipse, in addition, it is necessary to calculate lambana and nati and bimba of moon (angular diameter) also is different for the purpose of solar eclipse. These three will be specially discussed in this chapter. Verse 2 : Reason of lambana and nati At the end of amavasya, rasi etc of moon and sun are same, even then they are seen in same direction only at the time of mid-day. On other times, they are not in the line passing through centre and surface point of observation. Why this happens for times other than mid day, will be described in this chapter. When sun and moon are in mid point of sky, their direction from centre and surface of earth is same. Verses 3-6 - Meaning of lambana and nati Sphuta ending time of amavasya calculated from sphuta moon and sun is called samaparva Kala'. This time after lambana correction is the middle time of grasa (eclipse) in solar eclipse. This is sphuta amanta time for the place. At amanta time calculated from earth's centre, the difference between directions of sun and moon Solar Eclipse 547 is called lambana. This difference arising due to observation from earth's surface, and in east west direction is called lambana'. Its component in north south direction is called 'nati' or 'avanati'. When sun and moon are in mid sky, the line from earth centre to their centres passes through the surface point, hence there is no lambana or nati. When moon and vitribha lagna (lagna-90° on ecliptic) is same, there is no sphuta lambana, only nati is possible. When north kranti of vitribha lagna is same as (north) aksamsa of the place, then it has no nati also. When vitribha lagna' s north kranti is more than aksamsa of the place, moon (at vitribha lagna) has north nati. If north kranti of vitribha is less than north kranti of the place, or kranti is south, then moon has south nati. In amavasya (corrected with lambana), moon and sun have same rasi etc, hence nati in north south direction is easy to calculate. Verses 7-15 : Sphuta lambana by successive approximation - Instantaneous position of sun is found by method explained in sphutadhikara and from that, lagna of samaparva kala is calculated. By deducting 3 rasis (vitribha), again kranti is found for that. This kranti and aksamsa (direction of equator) being in different direction, difference is taken. They are added if they are in same direction. Result will be natamsa of vitribha lagna; On subs tr acting this from 90°, it gives unnatamsa. Jya of this unnatamsa is called sphuta drg-gati. Earth half diameter (800 yojana) assumed to be in sun or moon orbit, its angular diameter is 54# Siddhanta Darpana found in kala. (For sun's orbit, it is divided by 2213 and for moon's orbit multiplied by 6/85 according to verse 15 of previous chapter). Result is called 'Kuchanna Kala'. This is equal to the parama nati of sun and moon. Difference of these two is the parama (maximum) nati in solar eclipse. „ . . Daily motion of sun Parama nab of sun = 164 _, ... Daily motion of moon Parama nati of moon = - — 14 Difference of parama nati of moon and sun in vikala is divided by difference of daily motions of moon and sun in kala. Result in danda etc. will be parama lambana time. Parama lambana time in danda etc. is multi- ■ ■ plied by vitribha sanku of desired time and divided by trijya (3438). Result is antya of lambana. Jya of antya (in asu) is called para. Alternatively; sphuta drg gati (vitribha sanku) is multiplied by 100 and divided by 216. That will give the same para. Now jya of difference of vitribha lagna and sun is multiplied by para and divided by trijya. Result is lambana jya. Its arc in asu is sphuta lambana. If sun is west from vitribha lagna, this lambana time in asu is added to samaparva kala, otehrwise it is substracted. Result is sphuta samaparva kala. For this sphuta samaparva kala, we again calculate sphuta sun and vitribha lagna and lambana is calculated from their difference again. After repeated corrections, when there is no Solar Eclipse ^ 9 difference between two samaparva kala, that is the correct larhbana. Notes - (1) Approximate use of this method has already been made in verse 4 of previous chapter to find possibility of solar eclipse. First we derive the equation of parama nati (already explained in appendix to triprasnadhikara). Figure 1 - Parallax ol moon C is centre of earth and M is moon. From a local place O, the moon's zenith distance is z' and z is zenith distance from centre of earth. If OC = p, radius of earth for the place and CM = r, distance of moon from earth centre, then in AOCM sin COM sin OMC But Sin L COM = Sin (180 °-z') = Sin z' L. OMC = L 71 OM -L OCM =z'-zo = p i.e. parallax. Thus Sin p = {plr) sin z. Maximum parallax P= ^ occurs when z=90° ie.Sin z = 1. This is parallax when moon is at horizon radius of earth Thus parama lambana P = -^. . , ^« "/ r Distance of moon 550 Siddhanta Darpana This is angular radius of earth if it is kept in moon's orbit, hence it is called 'Kuchanna' ex- pressed in kala (minutes) i.e. ku = earth, channa = removed (to moon's orbit). Similarly parama lambana of sun is. earth's angular radius if it is viewed in sun's orbit. Alternative formula - For moon P in kala 3438 6 = radius, of earth x = — X radius of 4o/UO o5 earth (Verse 15 of previous chapter). But radius of earth = 800 yojana, moon's daily motion in kala is 790/35" which is slightly less than earth radius. Hence mon's daily motion P - if (2) Similarly parama lambana P' of sun is (mean value) n, ,. , , 3438 Earth radius kala F = radius of earth x — — — = 76,08,294 2213 (Result mentioned in verse 15 of previous chapter) = Sun daily moiton Earth radius x — — _ 2213 x sun daily motion mean Sin daily motion x 800 Sun daily motion 2213 x 59/8 = 164 ^ (2) Explaination of the terms : In figure 2, LNE is horizon and Z is zenith in celestial sphere. MVS is ecliptic and K its pole. Solar Eclipse 551 M is meridian point of ecliptic and V is vitribha lagna, i.e., shortest distance from Z. Since ZV is perpendicular, it bisects the ecliptic above horizon, hence V is at 90° from horizon point L called lagna. Thus it is called vitribha or 3 rasi less (than lagna). Figure 2 - Explanation of drgjya etc ZA is perpendicular to MK, so sin Z A is drg gati of madhya lagna M (or smaller drggati). ZB is perpendicular to SK - so that R sin ZB is drggati of sun S (or larger drggati). ZS is zenith distance of S; R sin z is drg jya. ZM is zenith distance of M, R sin ZM is madhya jya. Distance from Z in direciton of ecliptic is thus drg gati. Distance from Z in direction perpendicular to* ecliptic is drkksepa. Thus drkksepa of M, V and S are AM, ZV, SB. Total distance from z is drgjya. (R sin ZS) 2 = (R sin ZB) 2 + (R sin SB) 2 or drg jya 2 = drg gati 2 + drk ksepa 2 - - -( 4 ) (R sin SB) 2 = (R sin MA) 2 = (R sin ZM) 2 - (R Sin AZ) 2 or drk ksepa 2 = drgjya of madhyalagna 2 -drggati of madhyalagna 2 (5) 552 Siddhdnta Darpana (3) Lambana antara of Sun and moon z K Figure 3 - Lambana of solar eclipse Portion ZVSB of figure z is repeated here as ZVMA. M is the common geocentric position of moon or sun at end of amavasya. S' and M' are apparent positions of sun and moon due to parallax, when viewed from surface. Thus MM' = p, MS' = p' * VM is ecliptic and K its pole. V is vitribha lagna, Z is zenith. M'D. and S'D' are perpendicular on ecliptic. M'B and S'B' are perpendicular on KM produced Arc MD or M'B is lambana i..e parallax of moon along ecliptic. Similarly arc MD' or S'B' is lambana of Sun. Arc D'D is lambana of solar eclipse or difference of lambanas of moon and sun. ZA is perpendicular to KM. Then from similar triangles MBM' and ZAM we have r» • /«* „v R sin ZA X R sin MM' R sin (BM') = R sin ZM But BM' = MD = R sin BM' approx or R sin MD = MD = — tS -^ — R Sin z Solar Eclipse 553 = D?g S ati x P 3 * 3111 * 1 lambana P (6) Sin p as From (D P = ^ Similarly MD' = Drggati X P' (7) where P and P 1 are parama lambana of moon and sun. Thus DD' or lambanantara = MD-MD' = Drg gati x (P-F) from (6) and (7) = Drg gati x parama lambana antara (8) Parama lambana (antara) in time units is the time in covering that distance by moon. Relative speed of moon is moon gati - sun gati = m' kala Hence Parama lambana time Parama lambana kala , = day m' kala/day Parama lambana vikala = ghati m'kala B * Thus DD' in ghati = drg gati X parama lambana antara ghati (8a) (4) Vitribha Sanku and drggati - In figure 2 ZV = nati of vitribha lagna = nati of equator - kranti ■ - - - (9) In figure (3), KV = KM=90° In similar triangles KZA and KVM RsinKZ _ R sin ZA RsinKV = RsinVM Rsin KZ x R sin VM or Drg gati R sin ZA = —=: 554 Siddhdnta Darpana But R Sin KZ = R sin (90° - ZV) = vitribha sanku . R sin VM = ista sanku of sun or moon. tu t^_ Vitribha sanku X ista sanku Thus Drggan = ■- ( i 0a ) Radius vitribha sanku x Jya of vislesamsa Radius where VM = difference of sun and vitribha called vislesamsa From (8), laihbana Faram lambana x vitrib ha sanku = ^dh^s" xjyaof- vislesmsa (11) r» i , 56 x 60 Parma lambana = — — danda 731 56 kala is moon's parama laihbana, sun lambana is negligible, it is converted to vikala, 731 is difference of moon and sun gati 56 x 60 x 360 = — ^ asu = 1658 asu (taking 56/6/35 for 56) Parama lambana 1658 100 radius = 3438 = 216 ^ lla * Actually it comes 207, but after parallax in moon rise it is 216. (5) Summary of procedure - Natamsa of vitribha sanku is found from its kranti and aksamsa equation (9) 'Para' is calculated from 100/216 x vitribha sanku (11a) Parama lambana x vitribha san ku radius (U) Then from equation (11) Solar Eclipse 555 Lambana = Para x Jya of vislesamsa in asu Para x Jya o f vislesamsa . or Larhbanajya = — jj^ in kala For local place on surface, moon will be in same direction as sun before geocentric amanta when sun is in east. Because both move in east direction and in east half of sky moon appears further east due to parallax. In west sky, moon will be towards west, and it will reach sun's apparent positon towards east after lambana time. Lambana will change at new position, hence the procedure is repeated for further accuracy. Verse 16-22 : Accurate lambana in a single step. Now I tell the method to find accurate lambana in a single step. At samaparva kala, kotijya and bhuja jya of difference between sun and lagna is found. Square of (difference of para stated above and bhuja jya) and square of koti jya are added. Square root of sum is karna. Koti jya multiplied by para and divided by karna gives mean lambana time in asu. From this mean lambana, samaparva kala is corrected and drggati of that time is multiplied by madhyama lambana and divided by initial drggati (Here drggati means drg gati of tribhona lagna i.e. vitribha sanku). This is madhyama sphuta lambana, as stated by Bhaskara II. If this is more than madhyama lambana, their difference* in asu is squared, multiplied by madhya lambana. Result is added to madhya sphuta lambana. We take any of these - 1st sphuta lambana of Bhaskara or second sphuta lambana - multiply 556 Siddhanta Darpana it by mean gati difference of sun and moon and divide by (first sphuta gati of moon - sphuta sun gati.) By this Bhaskariya lambana becomes more sphuta. With this value of sphuta lambana in ghap, parva kala is corrected as before. Alternatively, at sphuta parva kala, kotijya of difference of sphuta sun and lagna is multiplied by 276 x drggati and divided by trijya. This gives lambana in pala. This is multiplied by difference of mean gati of sun and moon and divided by difference of sphuta gati. This will give difference of sphuta and samaparva kala. Notes (1) Bhaskara formula - v — >C vP e /Sy \i / G "V o^ A~7l Figure 4 - Bhaskara sphuta lambana E is earth, circle with centre E is sun orbit, circle with centre O is moon's orbit, deflected due to parallax. Vitribha lagna V and V of the orbits are in same direction from E, so that when sun is at V and moon at V, there is no lambana. Maximum lambana for this position of ecliptic is when distance from vitribha of sun and moon is 90°. This is OE. Directions OM and ES of sun Solar Eclipse 557 from their vitribha is same, hence these lines are parallel and equal. In parallelogram ESMO, SM also is parallel and equal to OE. Thus SM is equal to parama lambana or 'para' in short. SM // VE, hence is perpendicular to horizontal lines at G and A. Now in similar triangles SDM and EMG, EG x SM SD = EM R sin (s ~ v) x para or R sin (lambana) = — (1) * Kar na EM ^___ = V M G 2 + EG 2 = V (SG - SM) 2 + EG 2 = ^[R cos (S ~ V) - para] 2 + [R sin (s-V)] 2 •••• ( a ) drkksepa sanku x R si n 25-5fr- where para = — — - (b) We get maximum lambana when drkksepa sanku = R i.e. vitribha coincides with zenith and ecliptic is vertical. Then it is 1/14 of daily movement of moon which is 360° in angles. Thus max. lambana = 360714 = 25-5/7°. Putting values of (a) and (b) in (1) we get the formula. (2) Further corrections : Vitribha lagna is 90° from lagna by definition, hence V-L = 90 °, L = lagna. Then Sin (s ~ V) = Sin [(V-L) - (S-L)] = cos (S-L) and cos (S ~ V) = Sin (S-L) Further correction is based on sphuta gati difference of sun and moon as we had assumed 555 Siddhanta Darpana average gati in formula (b) above. Larhbana angle = madhya lambana X madhya gati diff . = sphuta lambana time x sphuta gati diff. This ratio is basis for further correction. Verses 23-39 : Nati correction in sara After finding mid point of eclipse by methods described above, we have to find sphuta sthiti ardha in which sara of moon is to be corrected by nati. For this; last sphuta gati of moon is found for eclipse purpose as explained in chapter 6. That will be multiplied by sphuta lambana in ghati and divided by 60. Quotient will be added or sub- stracted in sphuta sun of samaparva kala as lambana correction. Pata of moon (rahu or ketu) is corrected with digamsa phala (1/10 of sun mandaphala - chanpter 6) and is substracted from sphuta moon. From this difference (candra-rahu), sara is calculated. Then from sphuta sun (sayana) of that time, lagna and vitribha lagna of sphuta parva kala is found and their kranti is calculated. South sara is added to aksamsa (where equator is towards south) and north sara sub- stracted to get saraksa. Saraksa and vitribha kranti are added, if in same direction or substracted for different direc- tions. Result is nata (north south distance of moon from zenith). Jya of this arc is natajya. This is also called versine of madhya lagna or udayajya. Madhya jya is multiplied by udaya jya and divided by dyujya. Square of quotient and square Solar Eclipse ^ 559 of madhya jya are added. Square root of sum is drkksepa. Alternatively, vitribha lagna is assumed sun, and for that position kranti and cara are calculated. 15 ghati + cara = dinardha and its difference with vibribha lagna is natasu. From this vitribha natasu; drkjya is found through utkrama jya, cheda, ista hrti, and vibribha sanku as explained in seventh chapter, (verses 45-51) Arc of this drgjya is vitribha natamsa. When vitribha kranti is north of local aksamsa, then natamsa is north, otherwise it is ' south. This natamsa and Sara of moon will be added, if in same direction, otherwise difference taken. Jya of the resulting arc is drkksepa. Thus there are two types of drkksepa. Both are separately multiplied by moon gati in kala and divided by trijya. Results are added when manda kendra of moon is in six rasis starting from karka (90° to 270°) or substracted for other six rasis. By this, drkksepa becomeis sphuta. When grasa is less than 1 kala or more than 28 kala, then the second drkksepa is used which is corrected with vitribha natamsa. For grasa between 1 to 28 kala first drkksepa is used which is corrected with saraksa vitribha kranti (1/60 of total eclipse is one kala - when moon has sara, eclipse will be less than half and hence sara is used for correction). Difference of sun (21/38) and moon parma nati (56/28/13) i.e. (56/6/33) multiplied by drkksepa and divided by trijya gives sphuta nati. 560 Siddhdnta Darpam Also sphuta nati = ^ , , , Drkksepa Drkksepa - ' r 61 This is in direction of drkksepa. Sara of moon and sphuta nati are added if in same direction and difference is taken for opposite direction. Resulting direction will be direction of greater value of sara or nati. From that, grasa and sthiti ardha are calculated according to method stated in last chapter for lunar eclipse. When north sara of moon is more than the aksamsa of the place, aksamsa will be substracted from it. Difference will be north saraksa. When north kranti of vitribha is less than this saraksa but more than aksamsa, then aksamsa is sub- stracted from vitribha north kranti and result atided to saraksa gives north nata. Vitribha north kranti if less, is deducted from aksamsa and from remainder; saraksa is substracted to get south nata. If saraksa is more than | remainder, their difference will be north nata. Sum of vitribha south kranti and aksamsa if less than saraksa, their difference is north nata. These corrections are necessary for all places having more than 1° aksamsa. Thus in almost all places except equator region, natamsa is calculated from vitribha kranti corrected with saraksa; and drkksepa, madhya jya, nata in north south direction are calculated. Notes : (1) Moon's latitude from ecliptic depends upon its distance from pata (rahu or ketu). Its effective latitude for solar eclipse is latitude corrected for nati. Solar Eclipse 561 Now nati itself depdnds upon moon's distance from zenith towards south - consisting of two components - Distance of vitribha from zenith (it is only in north south direction - it is sum of aksamsa and kranti. Distance of moon from ecliptic - i.e. sara, Thus total distance in north south direction is algebraic sum of. Kranti of vitribha ± aksamsa of place ± sara- -(1) These are added if in same direction and substracted if in different direction. Total sara = sara±lambana - - (2) (2) Nati of moon : In figure 5, Z is zenith, V is central ecliptic point (Tribhona lagna), S the sun, S' apparent sun due to parallax, and S'A the perpendicular from S' on the ecliptic. Then from similar triangles SS'A and SZV, Figure 5 - Nati of moon S'A or sun's nati (approximately R sin S'A) _ RsinZVxR Sin SS' RsinSZ (3) Sun's drkksepa x R sin SS' *^m ,„^ ^^ „ 1 ■ ■ i — — — i.i -i ■ ' '■- * R sin SZ But R sin SS' parama nati of sun x R sin SZ " R - - (4) 562 Siddhdnta Darpana _ Earth's semi diameter in yojana x R R sin SZ Sun's mean distance in yojanas R Hence, Sun's nati Sun's drkksepa x Earth's semi diameter in yojanas Sun's mean distance in yojanas Suns drkksepa x sun's true distance in yojana Sun's mean distance in yojanas Earth's semi diameter in yojanas x Sun's true distance in yojanas Sun's drkksepa x Sun's manda karna in minutes R Earth's semi diameter in yojanas Sun's true distance in yojanas or, Sun nati Sun's true drkksepa x Earth's semi diameter in yojana Sun's true distance in yojanas. Similarly moon's nati = Moon's true drkksepa x Earth's semi diameter in yojanas Mon's true distance in yojanas Alternate formulas From (3) and (4) Sphuta nati (difference of sphuta nati of moon and sun) Diff . of parama nati x drkksepa Radius K } as drkksepa of sun and moon is same when they have same longitude after lambana correction, giving the values — c k . -« aw 56/28/ 13 - 21/38 Sphuta nati = drkksepa x Solar Eclipse 563 \ / (3) Complete procedure for sthitiardha - (a) First of all, calculate the time of geocentric conjunction (ganitagata or karanagata darsanta or amanta). Then calculate the lambana for that time and treating it as lambana for the time of apparent conjunction, obtain the time of apparent conjunc- tion by the formula - Time of apparent conjunction = Time of geocentric conjunction ± Lambana for the time of apparent conjunction (1) + or - sign being taken according as the, conjunction occurs to the west or east of the central ecliptic point. Next, calculate the lambana for the time of apparent conjunction obtained and then again apparent conjunction is calculated from formula (1). For the time of this second apparent conjunc- tion, lambana is calculated and again aparent conjunction is calculated (third) by formula (1). This process is repeated till lambana for the time of apparent conjunction is fixed. Applying this lambana in formula (1) we get the correct time of apparent conjunction. This is the time of spasta darsanta or spasta amanta, and also the time of middle of the eclipse. (b) Sparsika and mauksika sthiti ardhas - Calculate the semi diameters of the sun and moon and also moon's true latitude corrected for nati as explained in notes (1) and (2), for the time of apparent conjunction. This is almost equal to 564 Siddhanta Darpana moon's latitude at first contact time pi. If S and M are semi diameters of Sun and moon, d is difference between true daily motion of moon and sun in degrees - sparsika sthityardha = — * -f ^- ghatis - - (2) In practice, one uses the semi diameters of the sun and moon for the time of apparent conjunction, because, for the time of first contact, there is negligible change. Therefore, time of first contact = Time of apparent conjunction - sparsika sthityardha (3) Next, calculate the moon's true latitude for the time of first contact thus obtained; and then find the sparsika sthityardha by formula (2), then time of first contact by formula (3). Then calculate the moon's true latitude for the time of first contact (2nd value), then calculate the sparsika sthityardha by formula (2) and time of first contact by formula (3) again. Repeat this process unitil the sparsika sthityardha and the time of the first contact are fixed. The sthityardhas and vimardardhas which are thus obtained are called madhyama (or mean), because they are still uncorrected for lambana. (c) Lambana for times of apparent first contact and separation-Calculate the lambana for the time of first contact obtained above and treating it as the lambana for the time of apparent first contact, obtain the time of apparent first contact by the formula — Solar Eclipse 5*5 Time of apparent first contact = time of first contact ± lambana for the time of apparent first contact - - - - - (4) + or - sign being taken according as the first contact takes place to the west or east of the central ecliptic point. For the time of apparent first contact, thus obtained, calculate the lambana afresh and applying it in formula (4), obtain the time of first contact again. Repeat this process until the lambana for the time of apparent first contact is fixed. Similarly, find the lambanas for the times of apparent separation, immersion and emersion (d) Sparsika and mauksika sthityardhas, cor- rected for lambana — The madhyama sparsika and madhyama mauksika sthityardhas corrected for lambana, are called true (sphuta) sparsika and sphuta mauksika sthityardhas. They are obtained by the formula. True sparsika sthityardha = time of apparent conjunction - time of apparent first contact True mauksika sthityardha = Time of apparent separation - time of apparent conjunction. Similarly, True sparsika vimardardha = Time of apparent conjunction - time of apparent immersion True mauksika vimardardha = Time of ap- parent emersion - time of apparent conjunction Verses 40-42 : More accurate value of moon diameter (bimba) - Bimba (angular diameter) of 566 Siddhdnta Darpana sun and moon is calculated according to method given in previous chapter on candra grahana. Now method is being given to make it more accurate. This has not been told by any earlier scholar (acarya). If manda kendra of moon is in six rasis starting with karka, its koti phala is substracted from trijya, otherwise they are added. Square of the result is added to square of manda bhujaphala. Square root of sum is substracted from twice the trijya. By remainder, square of trijya is divided. Result will be manda karna in lipta (i.e. kala or minute of arc). Mean birhba kala of candra (31/20) is multiplied by trijya (3438) and product (107724) is divided by manda karna. It will give sphuta bimbamana of moon. Notes (1) If R and r are radius of main circle and manda paridhi, then Koti of karna = R + r cos when is manda Kendra Bhuja of karna = r Sin 6 Hence, karna K is given by K 2 = (R + r cos Of + (r sin 0) 2 (1) This is the correct formula. However, in place of bhuja phala or koti phala we take the lower value Bhujaphala = r sin x R/K Similarly, kotiphala, r cos also in reduced in same ratio. Thus we take K! 2 = R + r cos $ x — \ / 2 '' rsinfl x R x2 K Solar Eclipse 567 = R 2 + ^ (cos 6 + Sin 6) + ^ 2 K (cos 2 6 + Sin 2 0) rR or Ki = R + " (cos0 + Sin ^) approx. (2) 2 R-Ki = R - — (cos 6 + Sin ) XV R 2 R 2 2R - Ki rR R - — (cos + sin 6) is. R 1 - £ (cos + sin 6) = R [1 + -^ (cos 6 + sin (9)] = K from (2) (2) Mean bimba X mean distance (trijya) = True birhba x true distance (manda karna) = Diameter in length units. Verses 43-45 : Methods for calculating tamomana - (1) At the time of sphuta amanta time, we find sanku and drgjya from spasta sun. Parama lambana (56/28) is substracted from sanku. Squares of remainder and drgjya are added and of the sum, square root is taken. This will be tama karna (chaya karna). Sphuta candra birhba is multiplied by trijya and divided by tama-karna 568 Siddhanta Darpana This gives tamomana or grahaka (eclipser) value in solar eclipse. (2) Alternatively, 1/60 of sphuta candra bimba is multiplied by sanku of sphuta parva time and divided by trijya. Quotient is added to sphuta candra bimba to get tamomana. (3) Due to hard labour involved in calculating tamo bimba through sanku etc., I have found an easy method also for this. Unnata kala in ghati at the time of sphuta parva kala is multiplied by 2, the product in vikala is added to sphuta candra bimba in kala etc. Notes : (1) In previous verses sphuta candra bimba has been calculated for its variation in distance from earth's centre. However, due to parallax in observing moon from surface, its angle from vertical is increased, but distance is decreased. Though we correct the angle difference, the distance difference still remains. Since moon is seen at a nearer distance due to parallax, its effective angular diameter will appear increased. We have to calculate the increased bimba mana. Here tamo-mana is not the value of shadow, because shadow is not the cause of solar eclipse. Moon disc itself appears dark compared to sun and is called tama. (2) Derivation of formula - Figure 4 after verse 22 may be referred to For clarity, a smaller figure is made here (figure 5.) OZ is vertical and ZSH the great circle from Z through S, centre of sun and Moon. OH is horizon line. Sun and moon are at same place on samaparva kala, but figure at amanta time is shown when M is separate due to 569 Solar Eclipse parallax, so that distance difference is shown. iSOZ = Z is distance from vertical. SP is sanku = R cos Z, and SN = R sin Z is drgjya. Figure 5a - Tamomana increase in bimba due to parallax Due to parallax, moon is lowered to M' where SM' is equal to parama lambana of moon (as in figure 5a). SM' = P. On celestial sphere moon is seen at M in that direction. SM is small and this arc and straight line are almost same. In right angled triangle SMM',Z SMM' = 90°, L SM'M = z' when ZOM = z'. At sphuta samaparva kala z = z'. Thus lambana SM = SM' sin z' = P Sin z' which is according to the formula for lambana. It comfirms that apparent height of moon is lowered by distance SM' = parama lambana. Apparent distance from surface is OM' = tamo-karna of moon OM' 2 = OP 2 + PM' 2 = NS 2 + (SP-SM') 2 or tamokarna = V drgjya 2 + (Sanku - parama lambana) 2 " 0) Sphuta bimba of moon has been calculated for the distance of radius OM from earth's centre. / Trijya _ x Tama Kama parts of ^ Siddhdnta Darpana Apparent bimba at M' is bigger, which is tamomana. Hence, linear diameter being same linear diameter = tamomana X OM' = sphuta bimba X OM Sphuta bimba x Triiya or, tamo mana = — — — ^— o) Tama karna v ' (3) Alternate formula - Increase in sphuta bimba = bimba = OM - OM' _ M'M _ P cos z / OM' " OM' " R - P Cos z' (3) p. when z = O, at Z, increase is maximum = R- P Absolute in crease is P cos z' Then fractional increase in sphuta bimba _ P Rcosz' - R _ p ^ — / as P cos z' « P 56/28 Sanku J_ Sanku 3438 - 56/28 Trijya " 60 Trijya (4) In 15 ghati unnata kala, increase in moon bimba is 1/60 of sphuta bimba = J/60 x 30 kala approximately, when moon is at Z. 1 30 Hence in 1 ghati increase is — X — kala 60 15 = 2 vikala approximately Thus for each ghati unnata kala, sphuta bimba increases by about 2 vikala. Verses 46-47 : Hara of solar eclipse. Sphuta candra gati is multiplied by 1/60 of sphuta samku of sun and divided by trijya. In 571 Solar Eclipse auotient, final sphuta gati of moon is added. Sum substracted from sun gati will be hara at the time of eclipse (mid time). Unnata kala in ghati is reduced by its 1/8, remaining is assumed as kala and added to final sphuta gati of moon and sphuta gati of sun is substracted. Result is hara of sparsa and moksa time. Notes : (1) Hara means multiplier; here the purpose of this multiplier is not mentioned. However, in verses 46-50 of previous chapter on lunar eclipse, hara is used for calculating amount of grasa (magnitude of eclipse) at desired time. Hara in that context is difference of moon's speed and sun's speed. For solar eclipse this needs accurate calculation and correction for lambana. Hara = Candragati - surya gati (1) In this, variation due to parallax is only in candragati as the parallax of sun is negligible. linear d iameter Bimba = — ttt true distance linear motion ^ true distance Thus bimba and gati of moon both increase in same proportion due to apparent decrease in distance due to lambana or parallax. Thus according to first alternative formula in note (3) of previous verse, equation (4) is Proportional increase in candragati 1 sphuta sanku ,~\ 60 Trijya Siddhanta Darpana This correction put in equation (1) gives the first formula for lambana corrected hara (2) Hara for sparsa or moksa time - At 15 ghati unnata kala the increase in candragati from (2) is 1/60 part of its gati, . when moon is at Z approximately. This increase is 79035/60 = 13-1/6 kala ap- proximately. Hence, proportionate increase of each ghati in moon gati is 13 1 6 __ 11 ^ 1 15 ~ 1 ~6X15~ 1 "8 kala apprOX - Thus ghati is reduced by its 1/8 and remaining part taken as kala is the increase in daily motion of moon due to parallax. For this, unnata ghati of sparsa or moksa time is taken. Verses 48-49 : Difference in solar eclipse at each place. In lunar eclipse, shadow of earth and moon - both are at same place (in moon's orbit), hence grasa is same at all places, because there is no parallax. But in solar eclipse, chadya sun and chadaka moon are very far from each other. Only at a particular place, they may be in one line, but at other place they will be seen in different direction due to lambana (or parallax). Thus solar eclipse has different magnitudes for different places). Even due to a small difference in east west or north south direction, there will be difference in total eclipse, annular or partial eclipse. Hence, they are to be calculated separately for each place. Notes : Location of the point of observation is only reason for solar eclipse, other wise they Solar Eclipse 573 are vastly far from each other. This has been explained in beginning of previous chapter and while calculation of solar eclipse also. Solar eclipse is seen in a very small circle cut in moon's shadow cone by earth's surface. In north south direction from that circle, eclipse will become partial and then non existant. Due to relative motion of moon towards east the shadow circle on earth's surface moves from west to east and finally leaves. Thus the eclipse is earlier in west and later in eastern places on the strip of earth surface. Thus due to east west difference of places, eclipse times and grasa times will be different (according to standard time also). When tip of shadow cone is about to leave earth surface, before and after the strip, when circle on surface is of zero radius, extended shadow cone touches the surface. Then annular eclipse is seen at those places. Verses 50-53 : Madhya sphuta sthiti kala According to rules explained in candra grahana chapter, we calculate the sphuta sara, half sum of bimba. From hara of grahana time we calculate the sthiti ardha and marda ardha in ghati. By adding or substracting this from samaparva kala, we get times of sparsa, moksa, sammflana and unmilana. Then current lagna and vitribha lagna is found and lambana in east west direction is calculated. Sparsa and moksa times are corrected with this lambana. For these sparsa times etc, we calculate the lambana again and second value of sthitiardha and sparsa kala is found. For second values of 574 Siddhanta Darpana sparsa and moksa times, lambana is again calcu- lated and from that we get third value of sparsa or moksa. After repeated process, when there is no difference in successive values, we get the true values. Verses 54-56 : Sphuta sthiti kala by sara correction From difference of sphuta parva kala and these times of sparsa etc., we get the values of both sthiti ardha and marda ardha in ghati etc. Alternatively, we find the sphuta sara by single step method (verse 45 of previous chapter), and new values are found. From their ratio, sara for sparsa and moksa time is found. One difference is + ve and other is negative. Both changed by half the sum give the sara of sparsa, moksa time. From this sara, second value of second sthiti ardha is found. From that we find sara for sparsa, moksa and middle time sara. Then we find the difference of middle sara with the sara of sparsa and moksa times. By proportionate difference we again find sphuta sara ardha. After repeated process sthiti ardha becomes spasta. Notes : (1) Correction of sthti ardha for lambana by repeated process has already been explained after verse 53 and in notes after verse 39. (2) Suppose the sara at middle time be L and sparsa time sara is U. By single step method, the spasta sara is L'. Thus difference of sara is L'-l, in single step method and L-li in repeated method. Thus the difference of single' step method is to be changed by (L-li)/(Li-li) for correct difference. Thus we get accurate sara by one step method. If sparsa ■3 575 Solar Eclipse time sara is less than middle time sara, moksa time Sara will be more. Verse 57 : Method for small sthiti ardha We take the difference of sthiti ardha after 1st lambana correction and the sthiti ardha before that correction (initial value). Square of difference in pala is divided by initial sthiti ardha. Result is added to sthti ardha obtained initially. This process is done only for sthiti ardha less than 1 danda. From new values we get correct sparsa time etc. Note : Let the sparsa times counted from middle eclipse time be t , ti and t 2 before sara correction and after first and second sara correc- tions. For small sthiti ardha, second corrected time t 2 will be almost correct time. Change in sthiti ardha after 1st correction is / ti - to = to 1-t to same proportion \ It is assumed that sthiti ardha will change in / \ h to (ti - to) 2 - 1 t2"to = to to \ in next step also. 2 - Thus the correction is obtained by dividing square of difference of initial and first corrected sthiti ardha by initial sthiti ardha. Same process can be used for moksa time also. Proportional decrease or increase can be assumed only for small sthiti times. 576 Siddhanta Darpana Verses 58-60 - Single step method for sphuta sthiti time. We obtain sphuta sara for sparsa or moksa times after adding or substracting madhya sthiti ardha from lambana corrected amanta. If this sara is more than sum of semi diameter of the birhba; or equal to it, then madhya sthiti ardha is multiplied by grasa kala and made half. It is divided by difference of parva kala sara and sara at sparsa or moksa time (expressed in kala). By this, moksa and sthiti ardha are found in a single step only. From sthiti ardha times obtained, the corrected middle time gives sphuta lambana in one step only. Then sphuta sara will be found for lambana corrected sparsa and moksa times in one step only. Note (1) Grasa kala is amount of grasa expressed as ratio of diameter of eclipsed planet, out of total kala of 60. Thus sum of semi diameters - sara 8 Diameter of eclipsed graha When sara is more than semi diameter sum, then the planet will not be eclipsed and eclipse time will be shortened. Average value of sara between madhya kala and sparsa time is taken. When grasa is small, its value nearer to middle time is taken, as the real sthiti ardha itself is shortened. Verses 61-62 : Annular eclipse In solar eclipse when birhba of sun is more than tamo-bimba (apparent birhba of moon in- creased for parallax, then eclipse will be annular (valaya grasa). Then, from sum of semi diameters, diameter of moon is substracted. From square of A ;i Solar Eclipse 577 the difference, square of sphuta sara is substracted. From square root of this difference, we find sthiti ardha etc. in pala as per method described in lunar eclipse chapter. This sthiti ardha pala is corrected for larhbana and on adding or substracting from samaparva kala, we get beginning and end times of valaya grasa. Notes : This method is same as that of total eclipse time in which difference of semi diameter is taken. In this case, we get valaya grasa instead of total eclipse, because moon bimba is smaller. Verses 63-64 - Reason for extra methods Brahmagupta (son of Jisnugupta) had ob- served errors in the calculation of eclipse durations, hence in his Brahma-sphuta-siddhanta, stated at the end of tithi chapter, corrections for nadi (ayana drk karma), bhuja of nata, its jya etc. The method described by Bhaskaracarya in his Siddhanta Siromani also doesn't give correct eclipse duration. Hence, on the difficult topic of solar eclipse, I have stated many more things. Notes : Already many new improvements have been described to get more correct values of moon bimba etc. Now entirely new methods are being described for correct duration of eclipse. After that, modern methods will be described, as comments. Verses 65-72 t Eclipse duration through yasti - After calculating surya grahana by above rule, we multiply the sphuta Sara at the time of sparsa, middle and moksa, separately by the lagna kranti 578 Siddhanta Darpana jya of their respective times to give yasti for the three times. The three yastis are converted to para (1/60 vikala) and divided by hara (candra gati - surya gati) for the time of sparsa etc. When lagna kranti and sphuta sara are in different direction, this result in pala etc is added to time of sparsa etc otherwise substracted. Then true sparsa, madhya and moksa times are obtained. If this time is more than previous time (i.e. yasti -s- hara is added for different directions of lagna kranti), then it is the true time for sparsa etc. If new time is less then previous, it is multiplied by its lambana jya and divided by 'para' (stated arlier). Result is added to sparsa time etc., when sun is west from vitribha lagna, otherwise, it is substracted. This will give true times of sparsa, madhya and moksa. Madhya time will again be corrected with sphuta lambana to get correct value. Then squares of mid time sara and yasti arej added and square root of the sum is sphuta madhya kala sara. Then from the sara, sthiti ardha for sparsa and moksa are found. They arej separately multiplied by sphuta lagna dyujya for: madhya kala and divided by trijya. When sara of sparsa and moksa is in samei direction, first result is substracted from sparsa^ time and second result is added to moksa time. When the two sara are in different direction, reverse is done. The sparsa and moksa times are corrected for their lambanas to get true values. But sthityardha 579 Solar Eclipse is multiplied by dyujya of madhya kala and divided bv triiya. Lambana for parvanta is found from true sun of that time. At the time of sparsa and moksa, °ambana is calculated from position of moon at that time. Figure 6 - Sara correction through yasti Notes : To explain yasti, figure 10 after Triprasnadhikara verse 37 is "produced here. NZSZ'' is yamyottara vrtta, NS is horizontal line, ECE' is diameter of equator. QQ' is diameter of diurnal circle of sun and LI/ is diurnal circle of moon further removed from equator as kranti and sara are in opposite direction. These three circles are bisected by perpendicular PP' through poles - which is diameter of unmandala. BQ = Dyujya = semidiameter of diurnal circle = R cos <5 = corresponding to equator half day CE = 6 hours BD' = Kujya = Extra length of half day on diurnal circle = BC tan<l> H = R sin d tan 0> tjgQ Siddhanta Darpana CD = Carajya = Extra length of half day on equator in asu BD' = = R tan d tan# coso Now B is the position of sun when it has risen on equator. BD = height of sun at that time i.e. unmandala sanku. Height of planet above B is called yasti. Now A' is the joint position of sun and moon on ecliptic, A its position on equator corresponding to arc CA in asu. Let CA = K Height of A = CA cos <$> , where <S> is latitude) = K cos ® Height of A' above B i..e yasti of A' is = A'B cos O = K cos O cos d Its rate of increase with respect to angular distance from equator is - K cos O sin d Hence for change in distance corresponding to §ara s of moon, Increase in yasti = s k cos <I> Sin <5 s k cos <p sin d Proportionate increase * ^ cos< p = s sin d ---.- r - --(1) This yasti is the proportionate increase in time units of yasti and not the ista yasti meant in chapter Thus increase in yasti is equivalent to decrease in lambana, hence moon will reach the sparsa time after corresponding interval. Thus increase in sparsa time = yasti/hara or para where hara is Solar Eclipse 581 Relative speed of moon. When yasti is in vikala/60 and hara is in kala/day, the result is in day X 60 X 60 = in palas. Similar addition is to be made for J$he times of madhya and moksa also. When sara fis in same direction as kranti, substraction is to be ^jiade. When times are to be deducted they are Changed in ratio (lambana jya / sama mandala Iganku), because lambana jya is in time units. I Yasti is correction in sara of all times, hence leverage mid time sara is obtained by |§ara 2 +yasti 2 .) 1/2 tlferses 73-82 : Miscellaneous corrections I If among sparsika and mauksika saras, one is Nqual to middle time sara and other bigger, then llhere is a special method. I Ecliptic times are found by above methods land the sphuta sara of sparsa, madhya and moksa [Sime are multiplied by the kranti jya of lagna of Jtheir times and divided by trijya. When sara of j sparsa and moksa time are in same direction, these fresults are substracted from their sara> added if in | different directions. Result is multiplied by dif- |ference of sara and divided by 36. We get yasti in §!ipta. I This is multiplied by jya of distance between f&un and vitribha lagna and divided by trijya (3438), to get the third yasti. This third yasti in para is divided by sthiti ardha for sparsa etc and the result m pala etc is added to the times of sparsa etc. *vhen kranti and sara are in different directions, otherwise substracted. Thus we get the true times of sparsa, madhya and moksa. Madhya kala is I 15 <$i B&\. coo Siddhanta Darpana again corrected with sphuta lambana to get correct value. Then madhya kala sara and madhya yasti - both are squared, added and of the sum square root is taken. With this sphuta madhya kala sara, we calculate the sthiti ardha for moksa and sparsa limes. These are separately multiplied by dyujya of madhya kala lagna and divided by trijya. First result is substracted from sthiti ardha of sparsa and second is added to moksa sthiti ardha. Then both are corrected for their lambanas. When difference between spasta sara of madhya kala, and sum of semi diameters of birhba is more than 3 kala and kranti of sun is more than lagna kranti then surya grahana is calculated according to this method. If madhya sara is less than both the saras at sparsa and moksa time, more than both or equal to both, then first method should be used. Notes : (1) Kranti of sun is between the kranti of lagna and kranti of vitribha lagna, hence it to aproximated by either of them, which are at 90 from each other. No earlier astronomer had used kranti of lagna from whiich eclipse time can be calculated through yasti difference. Yasti differernce is same as difference of sanku. Both methods give same errors. In calculation with yasti one tune method has been used for calculating sthiti ardnas with sphuta sara corrected for yasti. (2) This method of yasti and previous methods are almost same. When grasa is 3 kala or more, Solar Eclipse 583 (difference of sara and sum of semi diameters), then the approximate distance between sparsa and moksa places will be (sun birhba + 3 kala) = 36 fcala aproximately. Hence sara difference is divided by 36 and resulting yasti is added to middle time jara. Approximately same will be added to other Saras also. Verses 83-85 Only that grahana (eclipse) is meaningful, which is seen from local place. No auspicious functions are needed for the grahana pot seen at a place. Thus lunar eclipse in day time :§t. solar eclipse in night time are not considered |is grahana for that place. But even at the time of part solar eclipse in liay time or part lunar eclipse in night should be Observed according to smrtis. Bath, charities etc should be done; cooking sleeping etc are prohibited. , As in lunar eclipse, in solar eclipse also grasa from time and time from grasa is calculated. Simlar method is used for aksa and ayana valana. Note (1) Amount of grasa and time in solar -eclipse. Let T be the Indian standard time of conjunc- tion in longitude, p is latitude of the moon, Pthe hourly change in latitude (north latitude and feotion towards the north being considered posi- tive), -M is excess of hourly motion of moon in tengitude over that of sun. L is angular radius of moon, S angular radius <?f Sun. Then at anytime t hours after conjunction, Ihe distance between the sun and moon's longitude ■' ■'.'*' ■•''■' ..^ mi 584 Siddhdnta Darpana is Mt and the moon's latitude is (p + Pt). So the distance between their centres is [M 2 t 2 + (p +. Pt) 2 ] 1/2 The eclipse begins or ends, when their rims appear to touch. This can happen, even if the distance between them is greater than L+S, for the moon's parallax may push it towards the sun. The maximum of this effect is II-H' (= II); II being the equatorial horizontal parallax of the moon, II' of sun which is negligible. Thus the rims can appear to touch when the distance between the centres is II + L+S (=d) at the most. Then [M¥ + (p+Pt) 2 ] = d 2 gives the times of the beginning and end of the general eclipse. Solving for t, we get -pP - * M 2 + p 2 + » 2 P 2 d 2 -p : M 2 + P 2 M 2 + P 2 Vi In this, the upper sign (-) is taken for the beginning, and lower for the end. T+t is the 1ST of the beginning or the end. At any given place, the eclipse begins or ends when the rims appear to touch at that place, i.e. when the apparent distance between centres is L+S. Now at any time T near the times of conjunction in longitude, let the apparent distance in longitude between the centres be m, the apparent excess of moon's hourly motion in longitude over the sun be M, apparent difference in latitude p, apparent excess of moon's hourly motion in latitude over that of sun be P, the sum of angular radii of sun and moon be d, and its variation per hour D. By apparent is meant here '(as affected by parallax)'. Solar Eclipse 5S5 Apparent m = real m + II cos A. Cos B (1 + II cos A Sin B) Apparent p = (real p+II Sin A) (1+H cos A Sin B) Apparent (L+S) = S+L (l+II cos A. Sin B) where A is the zenith distance of vitribha lagna given by Sin A = sin co cos <p sin v - cos to sm<p and B is (lagna - moon's longitude) given by B = Tan ' l [tan 1/2 (90° +v) cos 1/2 (90° + <p -w) / cos 1/2 (90°+ c/> +w)] ♦Tan" 1 [tan 1/2 (90°+v) Sin 1/2 (90°+ 0-w) Sin 1/2 (90°+ <p +w)] where ^ = latitude of the place to = obliquity of ecliptic (parama kranti) and v = sidereal time in degrees at the moment given by v = 97° 30' + east longitude of place in degrees from Greenwich + mean longitude of sun + 1ST at that moment in degrees. For strict accuracy, the geocentric latitude and horizontal parallax at that latitude should be used. If T is the time for which we have found m, p and d, the apparent distance between the centres of the sun and the moon at any time t hours after T is [m+Mt) 2 + (p+Pt) 2 ] 1/2 When this time is equal to d+Dt, the eclipse begins or ends. Thus eclipse begins or ends at 586 Siddhanta Darpana dD-mM-pP T + + M 2 + P 2 i Vi (mM + pP - dD) 2 d 2 - p 2 - m 2 (M 2 + P 2 ) 2 + M 2 + p 2 The middle of the eclipse i.e. the maximum dD - mM - pP eclipse occus at T + — ; — . r M 2 + P 2 The total eclipse begins or ends, when the rims apparently touch, the sun being within the moon. The distance between them at such time is (L-S), so by substituting for d in the above formula another d equal to (L-S), we can find the times of the beginning and end of the total phase. S may be greater than L, so that moon may be immersed in the sun, leaving a circle of light all around. This is called annular eclipse. Beginning or end of the annular eclipse is got by making D = S-L. (2) Bessel's method - for calculating solar eclipses - Bessel's method for calculating the circumstances of a solar eclipse as seen from a given place on the surface of earth consists in choosing a suitable system of axes, finding coor- dinates of the observer with respect to these axes and putting down in terms of these coordinates, the condition that the observer lies on the boundary of the penumbral cone at the beginning or end of the eclipse. All variable quantities in this condition are written in the form x +x't, where Xq is the value of the variable quantity at t = o and x 1 is the rate of change of the variable quantity. The origin of time is chosen near the middle of the eclipse so that t is small. The condition now Solar Eclipse 587 becomes a quadratic equation in t, solving which we know the beginning and the end of the eclipse. Besselian elements - Through the centre E of the earth draw a line paralled to the line joining the centres S, M of the sun and moon. Call this Z axis, its positive direction being on the side on which sun and moon are situated. Choose the y axis to lie in the plane determined by the z-axis and the axis EN of the earth, the positive direction of y axis making an acute angle with EN. Finally choose the x-axis to be perpendicular to the axis of y and z, its positive direction being towards the point of equator, which the earth's rotation is carrying from the positive side to the negative side. East Figure7 - Bessalian elements for solar eclipoe The plane z = O is called the fundamental plane. These axes are not fixed with with respect to the surface of the earth. Therefore, the coordinates of a point on the surface of earth keep changing. Certain quantities need to be calculated first which are required in the equations. These are called the Besselian elements. 588 Siddhanta Darpana (i) The elements d, x and y - Let the axes of x, y, z chosen as above meet the geocentric celestial sphere in X, Y, Z respectively. Let the right ascencion and declination of Z be (a,d) Then, as is evident from the figure, equatorial coordinates of X and Y are, (90 ° ■+ a, o) and (180 °+a, 90°-d). To find a and d, we note that x and y coordinates of the sun and the moon are same (for Z axis is parallel to SM) Let (d,<3) be the R.A. and declination of the sun and (di, d\) those of moon. If A is the sun's position on the celestial sphere, the values of Cos X A, Cos YA and Cos ZA can be easily written down. Thus, if (x, y, z) are the coordinates of the sun's centre S, and r is its distance from E, we have x = r cos XA =r cos <5 sin (a -a) y = r cos YA =r [Sin 6 cos d - cos d sin d. cos (ct-a)] z = r cos ZA = r [ sin <5 sin d + cos d cos d cos (a-a)] Similarly, coordinates (xi, yi, zi) of the moon are xi = ri cos 6 i sin ( a\ -a) yi = ri [Sin d\ cos d - cos diSin d cos («i-a] zi = ri [sin 6\ sin d + cos 6 cos d cos («i-a] where n is distance of moon's centre from E. Solving the equations obtained by putting x = xi and y = yi, we get a and d, the later being one of the Besselian elements. Substitution of these Solar Eclipse 589 values in the expressions for x and y will give us x and y, the other two elements. Values of x and y are calculated at the interval of 10 minutes for tlie whole duration of the eclipse. Therefore, x' and y', the variations in x and y per minute can *ilso be easily determined. The elements x and y are obviously the coordinates of the centre of the shadow on the fundamental plane. (ii) The element fi - Let fi be the hour angle of Z from the meridian of Greenwich at the instant. The Greenwich sidereal time is g. Since the R.A. of Z is a, the value of p is G-a. After ft has been tabulated at intervals of 10 minutes, fi' (the variation of u per minute) can also be easily tabulated. (iii) The elements ft, f 2 - The semi vertical angles of the penumbral and umbral cones are denoted by ft and fe respectively. Now the radii of the sun and moon are R and b, and the distance between their centres is approximately r-ri; so ft and f2 are given by • c R + b • < , R - b sin f x = , sin f 2 = ~ — 7 r - ri r - n (iv) The elements h, h - The radii of the circles in which the penumbral and umbral cones intersect the fundamental plane are denoted by h and h respectively. These also can fce fcmnd by simple geometry. Ii = b sec fi + zi tan ft and U = b sec k - zi tan h 590 Siddhanta Darpana where z a is the distance of the moon's centre from the fundamental plane and has been found in paragraph (i) above. In the Nautical Almahc, the quantities x, y, sin d, cos d, fi, li and h are tabulated at the intervals of 10 minutes for every solar eclipse. It is to be noted that these quantities relate to the whole of earth and not to any particular place on it. Circumstances of solar eclipse at a given place - Let p and O' be the geocentric distance and latitude of the place and x its longitude west of Greenwich . The hour angle of Z from the meridian of the place is fi-X since the hour angle of Z from Greenwich is j*. So if (§,17,?) are the coordinates of the place at any instant, we have £ = p cos <p' sin (a -A) 77 = p [Sin (p f cos d - cos <p' sin d cos (a -A)] % = p [sin <p' sin d + cos <t>' cos d cos (w - A)] The values of ( £17, £ ) can be computed for any instant. Also, since p is the only variable in these expressions, formulae for ( §', r\\ % ) (the rates of changes of £, t], and, £ per minute,) cart be found by differentiation and the numerical values of ( ?', rf, ?' ) can be determined for the time of eclipse. fundamental plane Figure B - Elements of solar eclipse Solar Eclipse 591 Consider now the sections of the penumbral and umbral cones by the plane z = £, i.e. the plane through the observer parallel to the fundamental plane. The sections will be circles; and if their radii are Li (for the penumbra) and L 2 (for the umbra), we have from the figure Li = li - 5 tan fi L 2 = k + £ tan f 2 from which Li and L 2 can be determined. Consider now the beginning or the end of a partial eclipse at the given place. At these two instants, the point (§ ,rj£ ) must be at the distance Li from the axis of the shadow, which cuts the fundamental plane in the point (xi,yi,o) and therefore cuts the plane z = £ in the point x,y,£. The condition for this is (x- £ ) 2 + (y-vf = L i 2 - - - 0) Replacing x, y, £ and rj by x +x'lt and similar expressions, (1) becomes a quadratic in t. Solving it, we have the times for beginning and end of the partial eclipse. It we write L 2 for Li in (1), we can similarly determine the beginning and end of the total eclipse. In (1) it is sufficient to take the value of Li or (L 2 ) at an estimated time close to the time of occurrence of the eclipse, for Li and L 2 change very slowly. To determine the point on sun's disc where the eclipse begins - Figure 9 represents the penumbra section by the fundamental plane. C is centre and CX', CY' are parallel to the axes of x and y. Then the generator of the penumbra thorugh Y' touches the sun in the most northerly point because the earth's axis lies in the plane 592 Siddhanta Darpana x= o. Also, the generator through X' touches the disc in the most easterly point. Suppose that ( £ rj £) lies on the generator through T. Then, if angle Y'CT = Li Sin 6 = (Xo+x't) - (go + g't ) Li cos 6 = (yo+y't) - too+^t) Substituting in it the values of t and the other quantities for the beginning or the end of the partial eclipse, we get the cor- responding value of 0, which is the position angle of the point where eclipse beginsFigure 9 - Starting point of solar eclipse or ends, because sun's disc is almost parallel to the fundamental plane. Verses 86-87 : Maximum and minimum values* of eclipse pala Maximum duration of candra grahana = 590 Maximum duration of total lunar eclipse i.e. marda kala = 273 pala Maximum duration of solar eclipse = 632 pala Maximum duration of annular eclipse (valaya grasa) = 48 pala Maximum duration of total annular eclipse (marda kala)*= 23 pala Maximum increase in duration of a tithi = 405 pala i.e. maximum value is (60+6/45) = 66/45 danda Solar Eclipse 593 Maximum value of naksatra tithi = 67/45 danda Minimum value of naksatra tithi = 52/12 danda • ■ Maximum increase in yoga (beyond 60 danda) = 162 pala Maximum decrease in yoga = 664 pala Maximum gati phala of moon = + 7742 vikala or - 4927 vikala. Maximum gati phala of sun = + 123 vikala or - 117 vikala Maximum sphuta lambana = 5/12 ghati Notes (1) Maximum duration of lunar eclipse- The total duration of a lunar eclipse is given in hours V , 2 \ a [D 2 - P 2 (1 - -2^-1)1 1/2 v p' 2 + m' 2 L p + m J where D is the distance between the centres of the moon and the shadow of first or last contact, P is the latitude of the moon at the time of opposition of the sun and the moon in longitude, p' is increase in P per hour and m' is the motion per hour in longitude of the moon, relative to the sun. This is clearly when D 2 = P 2 1 - ,2 P i.e. when D ± P ,2 P ,2 ,2 p + m / 1 " ,2 "2 V P +m , i.e. when P is numerically greater than D by Op' 2 / 2(p' 2 +m' 2 ) approximately. THis comes to about 14" on the average. Thus even when P is greater 594 Siddhdnta Darpana than D by upto 14", at conjunction, there can be eclipse. When P=D / duration of eclipse is not O, but 2Pp'/p' 2 +m' 2 , which is about 22 minutes. The duration is maximum, when latitude of the opposition P is o. It is equal to 2D/p' 2 +m' 2 But D, m and p, are function of 1 and 1', mean anomalies of moon and sun respectively. Therefore, the maximum duration itself varies between limits. Let 1 and 1' be anomalies at sthula parva; time of fictitious conjunction or opposition or opposition between. True moon = mean moon + 315' sin 1 True sun = mean sun + 127' sin 1' Equatorial horizontal parallax of moon II = 3447". 9 + 224".4 cos 1 for sun, W = 8.8" + 2" cos 1 Moon's semi diameter r = 939".6+61".l cos 1 Sun's semi diameter r' = 961".2+16".l cos 1 Radius of shadow S = 2545".4 + 228".9 cos 1 - 16".2 cos 1' V m ' 2 + p ' 2 = 1875". 6 + 260".l cos 1 - 5".0 cos r Now the distance between the centres of the moon and the shadow of first or last contact D = s+r - 3485". 0+290". cos 1 - 16.1" cos 1" 2D and j |2 T m + p 2 (3485".0 + 29(r\0 cosl - WW cosl') 1875.6 + 260.1 cos 1 ~ 5"0 co^l' . hi t.">. Solar Eclipse 595 This is a maximum when 1 = 1' = 180° and not when 1 = 1' = 0, as increase in denominator is more Thus maximum value is 2(3438-290+16.1) hours = about 238 minutes 1875.6-260.1+5 This is correctly given as 590 pala = 5/2 X 238 min. The lower limit occurs when 1 = 1'= and it is 2 (3438 + 290-16.1) ^ " . L ~~- — — — ~— — — r^ = about 212 minutes 1875.6 4- 260.1-5 If we do not neglect the function of 21, the maximum is about 237.4 minutes. Maximum duration of the total phase of a lunar eclipse is given by D = s-r. This also is maximum when sun andmoon are at opposition at Plhe nodes and when 1 = 1' = 180°. It is ! 2 (1605.8 - 167.8 + 16.4) r _ g |l 1875.6 - 260.1 + 5 I minutes I It is given in text as 273 pala = 0.4 X 273 = I 109.2 minutes ' ■ : |||2) Maximum duration of solar eclipse - I The formula for duration of a solar eclipse in ||gsneral on any place on earth (as opposed to the k " duration at any particular place) is the same as for &■■ /,;.■; fc- 'i: duration of a lunar eclipse. Only difference is that fe D *= H-H' + r+r' !*,'.*.: 596 Siddhanta Darpana and P is the latitude of moon at conjunction of the sun and moon in longitude. Here also the duration is not when p = D, but when P is numerically greater than D by about 20". When D = ± P, the duration is about 33 minutes The maximum duration of a general solar eclipse occurs when P = 0, i.e. when conjunction in longitude is at a node. It is given by 2D -7—3 it hours v p + m 2 (5339.9 + 285.5 cos 1 + 15.9 cos 1') = — ^ : hours 1875.6 + 260.1 cos 1 - 5 cos 1 This is maximum when 1 = 180° and l'=0 2 (5339.9 - 285.5 + 15.9) u Thus it is — hours inus it is 1875 6 _ 2601 _ 5 = 6 hours 18 minutes approx. = 378 X 5/2 pala = 945 pala Under this condition eclipse is annular. When 2 1 term is not neglected maximum is about 6 hours 16 minutes. The duration of a solar eclipse at a given place on the earth is given by — 3 jr~ corrected for & (P + m Y 2 parallax which changes rapidly and varies from place to place. But the maximum duration occurs when the central eclipse is at apparent noon. At this time, apparent semi diameter of moon is r + aobut 16". Also at noon, the retardation in relative hourly motion of moon is maximum, causing increase in duration of eclipse. For an hour angle Solar Eclipse 597 34° on both sides of noon, the average retardaiton is (850".3 + 55" .4 cos 1) per hour. Total duration is given by 2 (r + 16" + r') — rr- 2 7 — - hourly retardation due to v p + m parallax 2 (1917 + 61 cos 1 + 16 cos Y) (1875.6 + 260 cos 1 - 5 cos 1') - (850.3 + 55.4 cos 1) 2 (1917 + 61 cos 1 + 16 cos Y) 1025.3 + 204.7 cos 1 - 5 cos 1' When 1 = 180° and Y = 0, The maximum is about 4 hours 35 minutes = 275X5/2 pala = 687 pala (Text gives 632 pala) This occurs when conjunction occurs at a Node, central eclipse falls at noon, 1 = 180° and i'= The maximum duration of the annular or total phase at a given place is also at apparant noon lor the same reason. As the period is very short, *ve take the motion per minute. The duration of to annular eclipse near noon is given by *-.". 2 (r - r - 16) (31".3 + 4".3 cos 1) - (15" + 1" cos 1) K' i?.. w = 2 (5.7 - 61 cos 1 + 16 cos Y) 16.3 + 3.3 cos 1 This is maximum when 1 = 180°, l'=0 Thus it is about 13 minutes (34 pala approx) $gg Siddhdnta Darpana It is given 23 pala in the text. Minimum is clearly 0. The total phase is given by 2 (r +16" - Q (31".3 + 4.3 cosl) - (15" + 1" cos 1) _ 2 (61.1 cos 1 - 16.1 cos T - 5.7) 16.3 + 3.3 cos 1 This is max when 1 = 0, 1' = 180° when it is 2 X 71 ' 5 = about 7 minutes (= 17.5 pala) 19.6 This is not given in the text (3) Other limit : Other limits depend on the maximum and minimum values of speeds of moon and sun. First we change the maximum gati phala which is numdaparadhi x dainjka mean gat . Gati phala is + ve when manda paridhi is maximum at the end of odd quadrants. Hence maximum positive gati phala is more and negative gati phala is less. From this we get maximum and minimum gatis of sun and moon, by Max. gati = madhya gati + max. positive gati phala Minimum gati = madhya gati - maximum negative gati phala Minimum tithi = max (moon gat i - sun gati) Solar Eclipse 599 12 ° Maximum tithi = — : " ~tt iV "*~" nun moon gati - max sun gati 13" 20' Max yoga = mm (moon ^ + sun gati) 1 3° 20' Minimum yoga = ; ~T~ 7[r j & max ( mon g atl + sun gatl ) 13' 20' Maximum naksatra = min moon gati 13° 20' Minimum naksatra = — maximum moon gati Mean values of moon and sun gati are 790/35 and 59/8 Kala. There mandaparidhi at odd quad- rants is 12°6' and 31°30'. Mandaparidhi of sun at end of second quadrant is 12° 30' and at the end of 4th quadrant is 11° 54'. (Verses 95-96 of spastha dhikara, chapter 5). Moon's manda paridhi it is 188,5 kala more at end of 1st quadrant and minimum is 188.5 kala less at end of 3rd quadrant. Verses 88-89 : Prayer and conclusion The god is worshipped in forms of Parvati, Surya, Siva and Ganesa and gives fortunes to devotees. He also changes moon into rahu (at the time of solar eclipse) and puts little knowing earthly creatures in confusion by covering sun like a flake of cloud. The same god may remove our troubles. Thus ends the ninth chapter describing solar eclipse in Siddhanta Darpana written for tally in observation and calculation and education of students by Sri Candrasekhara born in renowned royal family of Orissa. Chapter - 10 PARILEKHA (Parilekha Varnana) Verse 1 : Scope - To show the direction of sparsa, madhya and moksa in surya and candra grahana clearly through diagrams, I explain the methods now. Verse 2-3 : Valana - An oblique ray of light bends in water but doesn't bend in vertical direction. Due to that reason, the size of sun and^ moon and sara of moon, remaining same, it looks- smaller in middle sky and bigger at horizon. Hence, earlier astronomers, changed the values of moon, sun earth's shadow in meridian depending on hara at that time. This is being explained now. Notes : Valana means bending. Light rays bend due to refraction, hence it is now called refraction effect. In appendix to Triprasnadhikara (chapter 7) this has been explained. If angle of incidence of light to a denser medium be i and angle of reflection be r then u = = a constant for the medium (1) sin r Hence the bending (i-r) increase with increase in angle of incidence. This angle is measured from perpendicular to the surface, hence in vertical direction there is no. bending. As we move towards horizon, the bending is more. Thus, at sunrise (or Parilekha 601 setting time) its lower end is at horizon having 90* natamsa and upper end has slightly less (90° -32') natamsa. Thus the lower end will be raised more compared to upper and it will be flattened and look more elliptical. The angle of bending or valana, R is R = K tan z' (2) where K = fi-\ and z' is apparent zenith distance Difference z-z' is proportional to K sec 2 z. dz = 32' (K sec 2 z) for sun. Hence apparent angular diameter is difference between apparent natamsa z and z' of upper and lower. Maximum refraction at horizon is about 35'. Its variation is very fast near horizon due to very high value of sec z near z = 90°. Natamsa Refraction K 0° 0" '<i ; ■ 5° 5" 10° 10" V; . 15° 16" a- ■ 30° 34" I. 45° 58" to-- 60° 1'41" mi Ml ■-■ ■ r 80° 85° 5'19" 9'51" Bns'-' 88° 18'16" K..* 88°40' 22'23" 90° 35' Thus apparent reduction in vertical angular diameter at horizon is about 5'. 5Q2 Siddhanta Darpana If D is the average of two perpendicular angular diameters observed at vertical distance z, then the real diameter is D [1 + 1/2 K (1 + sec 2 z] (4) which is bigger than the observed. This means that observed diameter will decrease as Z increases and is minimum at horizon. Verses 4-5 : Value of angular measure for birhba. Unnata kala sanku of moon (for lunar eclipse) or sun (for solar eclipse) is calculated for middle time of eclipse. We add 10314 and divide the sum by trijya (3438) to get the hara or value of 1 angula in kala. On dividing the bimba of planets or shadow or sara of moon by this hara, we get their diameters is angula units. Alternatively, half day is multiplied by 3 and added to unnata, kala of moon (or sun) and divided by half day to get the same hara. Value of birhba and sara in angula units is obtained by dividing their values in kala by this hara. Alternatively, for rough calculation, bimba Kala is divided by 3 to get its value in angula. Notes : (1) Surya siddhanta assumes (Candra grahana verse 26) that the proportional angular diameter of a graha is 3 units at horizon, then it becomes 4 unit at vertical position i.e. increase in the ratio of 4/3. Bhaskaracarya and Lalla have assumed 2-1/2 : 3-1/2 increase i.e., in ratio of 7/5. Actual increase as we have seen after verse 3 is from (32'-5') to 32' in sun's bimba i.e. in ratio of 32/27 = 1.2 approx. Thus the ratios 1.33 of surya Parilekha 603 siddhanta and 1.4 of Bhaskara II are much higher than the true ratio. Another approximation is that the increase has been assumed proportional to the angular rise above horizon upto value of 90° rise to top position, where it is maximum. Angle of rise 0° = 90°-z. Putting it in equation (4) above, apparent diameter is T D = ~ 1 + - K (1 + cosec 2 6) 2 v For = 0, lower term cosec °o = which is not correct approximation. However, the increase is in proportion to value of cosec and not proportional to as assumed. This is increase of average diameter. Vertical diameter will increase at double rate. (2) 1 angula = 3 Kala at horizon and = 4 Kala at vertical position Height is proportional to unnata sanku, as assumed. For height of R (Trijya = 3438') increase is 1 kala, U increase is — Kala Thus 1 angula = 3 + — Kala _3R±U Kal5 = 3 X 3438 + U^ R ^^ 3438 U + 10314 _, _ = — Kala 3438 604 Siddhanta Darpana This is the first formula Roughly half day is of 15 ghati when sun reaches at top. Actually it is still slightly away from zenith but that distance is ignored. Unnata kala is in proportion to half day taken as 90° or 15 ghati. Unnata kala LI Hence ' half day = ¥ U Unnata Kala or, langula = 3 + - = 3+ ^ ^ y 3 x half day + unnata kala half day This is alternative formula If we totally ignore the variation due to refraction, except for horizon position, diameter is almost same, and 1 angula = 3 kala is uniformly assumed. bimba in Kala . . . Thus =r~rr. — ~ r~ = bimba in angula Kala in 1 angula Verses 6-14 : Diagram for direction of eclipse On a ground, plane like water level, a circle of 57/18 angula semi-diameter is drawn with a compass. This is known as khagola vrtta having two valanas. From this centre only, another circle with radius of sum of semi diameters is also drawn which is called samasa vrtta. From same centre a third circle is drawn with radius equal to the grahya bimba (which is eclipsed) Now according to method explained in Triprasnadhikara north south line and east west lines are drawn in khagola vrtta. In lunar eclipse; Parilekha 6°5 sparsa is from east and moksa is in west direction. But in solar eclipse sparsa (beginning) is from west and moksa is in east direction. In khagola vrtta we mark a point at a distance from east point for lunar eclipse equal to jya of sphuta valana and in same direction as valana. A line from centre to that point is drawn. Similarly, at a distance from west point equal to and in direction of moksa time valana, another point is chosen and a line from centre is drawn. In solar eclipse, the order of valana lines is reverse i.e. sparsa in west and moksa in east direction. These lines are called valanagra rekha. Valanagra rekha cuts samasa vrtta on valana points. From these points, we mark the distance equal to sphuta sara jya of moon at the time of sparsa or moksa. These are called saragra vindu (in east for sparsa and nimHana and west for unmllana and moksa in lunar eclipse, opposite direction in solar eclipse). The line from centre to saragra point cuts grahya and moksa. Here sara and valana are given according to their current values. Sara is in north south direction, some times in angle direction like agni kona (north east). Notes : (1) Radius of khagola vrtta is 57/18 angula because 57° 18' = 3438' = length of radius. Hence 1/60 angula on radius or circumference is equal to 1 minute or kala. The method is same as in surya siddhanta, but there the radius is 49 angula where 1 angula was 70'. Radius of samasa vrtta or grahya vrtta will be calculated according to value of angula in kala 606 Siddhanta Darpana calculated in verses 4-5. Roughly 1 angula = 3 kala. Similarly length of sara also is calculated in angula. However, valana is measured on khagola vrtta where 1 pratyangula (1/60 angula) is equal to 1 kala or 1 angula = 1°. With this unit we measure the lengths. (2) Method of drawing is best explained by actual diagram. Figure 1 - Diagram for sparsa and moksa in eclipse ENWS - Khagola circle, 1 angula = 1°, 57/18 radius, E'N'W'S' - samasa vrtta, E,E' east points, N, N', North points; W, W west points S,S' South points, AB is grahya birhba EVi = Valana jya for sparsa in solar, moksa in lunar eclipse. WV2 = Valana jya for moksa in solar eclipse, sparsa in innar eclipse Vi'S = Current sara of moon Vi'S' current sara Vi', Vz are lines on samasa vrtta cut by OVi, OV 2 . OS; OS' cut grahya on A,B which are points of contact. Parilekha 607 Verse 15-30 : Further details for periods within sparsa and moksa. Now, I describe the details of eclipse between the end points of sparsa and moksa. In lunar eclipse, when moon is near rahu or ketu, spasta valana in khavrtta is given in own direction from east or west point in north or south direction. From these valana end points, we give two points at distance of 5 angula, in north direction from east valana, and south direction from west valana point. We draw a line through these points which also passes through centre of the circle. In solar eclipse, we mark a point from eastern valana point at a distance equal to lagna kranti in the direction of kranti. This point is joined with centre and extended to make it diameter. Sara of moon is put in perpendicular direction on its line according to direction of the sara. (Sara will be at.\ central point for middle position of the eclipse or any other point according to time of eclipse). From end point of sara a circle is drawn with radius of grahaka bimba (eclipser) (This circle is drawn in lunar eclipse on 5° difference line). The portion cut by grahaka bimba will be the extent of eclipse visible to people. Sara of sparsa, madhya and moksa periods are put at their positions. From the three end points of sara, we draw three circles with radius egiial to 1/3 of the distance between sparsa and From intersection of adjacent triangles two like figures are formed. The head tail lines of £hese fish figures join at a point which is centre 608 Siddhdnta Darpana of circle passing through these points. With this centre an arc is drawn through sara ends of sparsa, madhya and moksa which is the grahaka marga (path of the eclipsing planet or shadow). From centre of this grahaka marga, we draw a line in the direction of sparsa (eastern direction in lunar eclipse and west in solar eclipse), at the distance of grahaka diameter from sparsa point, there will be nimflana point on the grahya circle. Similarly, unmilana point on grahya circle will be on the moksa side of the grahaka marga. To find the amount of grasa at desired time we assume two parts of grahaka marga - from mid point to sparsa, it will be sparsa khanda and the other side will be moksa khanda. Their length is measured in angulas. The angula measure is multiplied by required time (after sparsa or before moksa) and divided by its sthiti ardha time. We give. a point at a distance equal to angula measure of required time from sparsa or moksa point. From that point, we draw a circle with radius of grahaka circle. The portion cut by this circle in the grahya birhba will be the required amount of grasa at desired time. Sum of semi diameters of grahya and grahaka is substracted from the required grasa in angula. A pointer equal to remaining length in angula is taken. With this, we find two points on grahaka marga at distance of grasa from centre of grahya circle. One point is in sparsa khanda and the other in moksa khanda. From these points we draw circle with radius of grahaka birhba. The portion covered by this circle will be the portion eclipsed. Parilekha 609 At the distance of difference of semi-diameters of grahya and grahaka from centre of grahya birhba, we get two points on grahaka marga - one on moksa khanda and the other on sparsa khanda. These are the points of nimilana (on sparsa khanda) and unmilana. Notes : Figure 2 - Diagram for sparsa and moksa in eclipse ENWS are direction points on khagola circle, 1° = 1 angula; Radius = 3438' = 57° 18' = 57/18 ahgula. E'N'W'S' - Samasa circle direction points, radius equal to sum of semi diameters of grahya and grahaka. For bimba and sara length, 1 angula unnata sanku kala = 3 + --— Tnjya Vi, V2 are valana points. EVi and WV2 are ^qual to magnitude of direction of valana jya. V1L1 = V2L2 = 5* i.e. 5 angula on Khagola circle which is equal to inclination of moon's orbit with ecliptic. Thus Li L 2 is path of moon for lunar eclipse. g20 Siddhdnta Darpana For solar eclipse V1L1 = V 2 L 2 = kranti of lagna. On its intersection with samasa circle and at centre, sara lengths at sparsa, madhya and moksa points are drawn, perpendicular to it. It will be least at the centre and in direction of sara at all places. Their ends are Si, S 2 , S 3 : The three circles through these points from two fish figures which intersect at point C. From C as centre with radius CSi = CS 2 = CS 3 we draw a circle. Si S 2 S3 arc is the grahaka marga on which eclipser planet or shadow moves. For lunar eclipse S 2 S3 is sparsa khanda and Si S 2 is moksa khanda. For solar eclipse Si S 2 is sparsa khanda and S 2 S3 is moksa khanda. Nimflana point P for lunar eclipse (or un- milana point for solar eclipse) is on grahaka marga such that S3 P = diameter of grahaka birhba. Length on grahaka marga is proportional to time. Hence for any point P Length Desired time S 2 S 3 " Sthiti ardha This formula is used to calculate grasa at desired time. Verse 30-35 : Another method of diagram - At saragra point on one side of valanagara rekha (sara is madhya sara), another line parallel to valana rekha is drawn. From its end points on khagola circle, a ppint is given towards north for lunar eclipse (south for solar eclipse) at a distance of 1/60 of Jya of local aksarhsa. Parilekha 611 From these two points and the point of madhya sara point (i.e. mid point of parallel line to valana rekha) we draw a circle as explained in above verse. Portion of this circle within samasa circle will be grahaka marga. On this path, we can find nimilana and unmllana points from centre of grahya circle at distance of difference of semi-diameters of grahya-grahaka, . as before. In this diagram sparsa and nimilana of solar eclipse can be seen in west direction and, for lunar eclipse in opposite direction very easily. This method doesn't need sara or valana time at time of sparsa, etc. But,, for diagram of solar eclipse, 1/3 of sara of moon (i.e. angula value) is kept at two places. At one place it is multiplied by sun sanku of that time and divided by 4400. Quotient is added at first place. On a single board both solar and lunar eclipses can be shown. Only difference will be that the direction of sparsa, moksa etc will be opposite for the two types of eclipses. Note : This is almost same procedure. In stead of marking sara at sparsa, moksa and mid points, we mark the middle sara only. In stead of other sara, we mark the kranti of lagna on khagola at distance from middle sara. Reason is that the diurnal circle of moon will be parallel to ecliptic and at same angular distance from lagna point of ecliptic as on middle point of eclipse. For solar eclipse sara is corrected for parallax. The correction is slightly less, which appears to g22 Siddhanta Darpana compensate effective increase of tamo-mana of moon as explained in chapter 9 verses 43-45. Figure 1 and 2 show, that both the diagrams for solar and lunar eclipse can be combined, which has been prescribed here. Verses 37-38 : Prayer and conclusion I pray to lord Jagannatha, who smiles with beautiful lips, beauty of whose round eyes defeats the beauty of morning sun of spring time and full moon of winter night, who gives freedom from fear to people flocking to Nilacala from different regions, and whose sight can emancipate the world. Thus ends the tenth chapter on diagrams in siddhanta darpana wirtten for calculation according to observation and instruction to students by Sri Candrasekhara, born in famous royal family of Orissa. '■■«'*' Chapter - 11 CONJUNCTION OF PLANETS GRAHA YUTI VARNANA (Conjunction of planets) Verses 1-2-Scope - While the planets are moving in their own orbits, their position is seen same from earth. This is called graha yuti (con- junction of planets). Graha yuti and its good or bad results are described in this chapter. According to Surya siddhanta, when tara graha (mangala etc.) are seen joint, then their (apparent) . coming together is called graha yuti or yuddha. When any tara graha comes together with moon, it is called samagama. When tara graha is with sun, it is not visible due to bright rays of sun, and it is called 'asta mita' (heliacal setting of planets). Notes (1) Planets do not really come together. They are in their own orbit which are far from each other. But due to parallax, they are seen together, as in solar eclipse, sun and moon are seen in same direction. However, the parallax is same for all positions from earth due to large distances of star like planets (tara graha). Compared % eclipse of sun, the diameters of tara graha are ^uch smaller and their orbits are farther and r, hence their conjunctions are rare. However, 624 Siddhanta Darpana their number is more causing different combina- tions of yuti and their sara also is small compared to moon's orbit, so we are able to see the yuti some times. (2) Moon is considered the king of stars and the naksatras as its wives. It lives with one naksatra each day like a husband and wife - 'naksate' means lives together. Thus conjunction of moon with any naksatra or tara graha is called samagama or happy union. Conjunction between tara graha is called 'yuddha, as it is not considered friendly. In this 'yuddha ' or war, the planet which is behind is like a chaser and takes away half the strength of the other planet which is considered defeated. This strength is considered in astrology for considering their power in giving good or bad results. The reduction or increase of strength is according to their mutual covering and depends on their angular diameters. At present, we follow the method of Sripati for calculating the reduction or increase in strength due to planetary war. Due to nearness with sun, the planets are invisible and called set due to sun. This has already been mentioned in chapter 6 and will be discussed in an independent chapter on it. (3) Varahamihira in his Brhat samhita, ex- plained in detail the various results of graha yuti. According to the degree of their seeming approach- ment, there are four kinds of wars (among planets) as stated by Parasara and other sages - Bheda (occulation or cleaving), Ullekha (grazing), Arhsu mardana (clashing of rays) and Apasavya (passing south ward). Ifrrijunction of Planets 615 Yeises 3-5 : Principles of computation We find rasi, amsa and kala of two planets in conjunction. When they are equal in ecliptic /kadamba prota vrtta), their values on equator are found (dhruva prota vrtta). From this, their sara and lambana are found. Then bimba (angular diameter) is calculated. In surya siddhanta - When faster of the two planets has greater longitude (i.e. it is towards east), then conjunction has already occurred. If it i*fe less (i.e. in west), then the conjunction is yet % occur. If both are vakri (retrograde) then reverse *oll happen, i.e. planet in east indicates, conjunc- tion is to occur, in west means conjunction has already passed. If one body alone is retrograde laid its longitude is greater (in east), then the conjunction is to come, if less, it has passed. Notes : (1) Conjunction is calculated first in longitude measured along ecliptic, when their positions are same. However, their difference in perpendicular direction (sara) and apparent devia- tion due to observing from earth will depend on position with respect to equator. Size of the bimba of planets will decide, at what distance they will meet. (2) Finding conjunction, time, whether gone or yet to come is very easy to find, from diagram. ft Figure 1 - Conjunction of planets 616 Siddhanta Darpana In figure 1, M is position on ecliptic, which is mesa 0° from which position of planet is measured. When arrow direction indicates rotation in east direction, the rasi arhsa etc (longitude) of two points A and B are MA and MB. When longitude of B is more, it is east from A as seen from figure.. When B is faster, it will move further east from A, and at some earlier time it was with A i.e. in conjunction. If A or western planet is faster it will meet B in time needed to cover AB with relative speed. If B is retrograde in east position A and B, both approach each other with their speeds, hence it will approach with speed equal to sum of speeds. When both are moving in western direction, obviously the reverse of direct motion will happen. Verses 6-9 : Finding the time and place of conjunction - At required time we find the bhogamsa (longitude) of the two graha and convert their difference into kala. This is separately multiplied by daily speeds of graha in kala. Each product is divided by difference of speeds if both have direct or both retrograde motion. But if one graha is margi and the other, vakri, the products are divided by sum of speeds in kala. If both planets have already joined and both are margi, then each quotient is deducted from the bhogamsa of its planet by whose speed it had been multiplied. If conjunction is yet to happen, then the quotients are added. If both are retrograde (vakri), reverse is done. If one is vakri and the other margi, then addition and substractipn are done as per rules explained earlier. By this, we get the bhogamsa of Conjunction of Planets 617 Kranti vrtta (position on ecliptic) where conjunction w has happened. If the kala of planets doesn't become j equal in a single operation, this process is repeated I again. Notes : In figure 1, longitude of A is MA, B is MB Differnece in longitudes is MB-MA = AB Speed of A is a and B is b kala per day Difference in speeds is a-b if a > b, then A will catch up with B in time AB / (a-b) if a, < b then B has gone ahead this difference I AB in time AB / (b-a) Thus in first case the longitude of conjunction AB for A will increase X a, increase in B will a-b be x b. This increase in A will be (a-b) a-b a-b i. " L more i.e. AB more and they will catch up. If a < b, then the conjunction time is earlier I and longitude of A and B will be reduced by | distances travelled by them. For retrograde planets obviously situation will I be reversed. If B is faster, it will catch up distance g . BA in time t = BA / (b-a) in which the longitude of B and A will be reduced by t b and t a. Suppose A is retrograde and B is forward motion. There relative speed is at a+b and their distance is increasing. Then they are together at time AB / (a+b) = t before the present time. In this time longitude of A was more by ta because 5ig Siddhdnta Darpana it is retrograde and B was tb less, in earlier time of conjunction. If A is direct and B is retrograde, then the plenets are approaching each other with velocity a+b and they will cover the distance AB in time AB / (a+b) = t when they will be together. After that time position of A will be ta more and of B will be tb less because it is moving in reverse direction. (2) We are assuming uniform motion of planets in the interval AB. Within this the speeds will change, forward motion may become retrograde and vice verse. Thus after getting the conjunction time approximately on basis of present speeds, we again calculate the position difference at this approximate conjunction time. Then we calculate more accurately as to when conjunction had occurred or would occur. Verses 10-11 : Sara of planets (From Surya siddhanta - Spastadhikara verse 56-57). In pata of mangala, sani and guru, correction for second sighraphala is made in same manner, in which it is done for the planet (i.e. positive result is added and negative substracted). This will give the true postions of pata of these three planets. But in pata of budha and sukra, correction is made with second mandaphala (used in third step of correction) in reverse manner - i.e. positive result is substracted and negative added. By this, true pata of budha and sukra will be known. Conjunction of Planets 6W From true postions of mangala, sani and guru, true positions of their pata are deducted to get viksepa kendra. Viksepa kendra of budha and gukra are found by substracting their true pata from their slghrocca positions. Jya of viksepa kendra is multiplied by madhya viksepa and divided by fourth slghra karna to get the sphuta sara. Notes : (1) Mean inclinations (viksepa) of planetary orbits - This has been explained by Bhaskaracarya II. in his chapter on grahacchayadhikara (siddhanta siromani). Reasons | of the method have also been explained. The values of madhya viksepa are given in chapter 5 - spastadhikara verses 28-33, reproduced here Planet Siddhanta Darpana Modern value value Candra 5'9' 5'8'42" Mangala 1*51' 1"51'<T Budha 2°44' 7°0'14" Guru ris- ri8'2r Sukra V 28' 3°23'39' Sani 2"29' 2 e 29'25" The values of superior planets are almost same as modern values. Bhaskara says that these values are for that time when slghra anomaly is equal to 90° + 1/2 R sin _1 a, where a is R sine of the maximum slghra phala. This is quite correct, // iff 620 Siddkanta Darpana E "A' Figure 4 Figure 5 ' because when the sighra anomaly has this value, the true planet is at point of intersection (P) of the deferent and eccentric circle. Then the planet is equidistant from Ei and E2 (figure 2) For superior planets, Ei is taken as earth's centre and E2 is sun, the mean latitude of the planet observed will be same, whether observed from earth or sun. Hence, maximum latitudes of the superior planets are same for geocentric and heliocentric observations. These are the mean values. For inferior planets, mean planet in this case is taken to be sun, the linear values of the latitude observed from E and S, the centres of Earth and sun will be in ratio SP/ES (figure 3) For mercury Conjunction of Planets $21 this ratio is 4/10 and for venus it is 7/10. Hence 420 X 4 the modern values are reduced to and 10 204 x 7 . - — — i.e. 168 and 142 which are approximately equal with the values given in siddhanta. (2) Pata is calculated for orbit round sun and converted to geocentric position - Figure 4 shows an inferior planet indicated by P and Figure 5 an superior planet J. Position of earth and sun are E and S., position of mesa 0° from sun and earth are A and A'. Position of node from sun and earth is N and N'. True position of inferior planet is P and superior planet is J. U is the mandocca position (i.e. sun) for P. Pata of inferior planet - Convex angle ASN is heliocentric longitude of node measured negatively, as node has a negative motion on ecliptic. Rule says that heliocentric sighra anomaly is added to this which Convex angle ASN + ZUSP = 360° + ZNSP - Z.ASU = ZNSP~Z.ASU Now longitude of planets is added here i.e. fUASU (= ^A'ES) p Result is Z.NSP. pi,-. ;> . <*.* , . RSinNSP x B bara as seen from sun is — — R P is maximum sara (latitude). I 522 Siddhanta Darpana As seen from earth this is to be reduced in ratio R/K where K is distance from earth i.e. sighrakarna. Thus sara seen from earth RSi " NSP X g which is the formula. K Pata of superior planet - True geocentric longitude of J is ^A'EJ = Z.ASJ' Substracting slghraphala EJS = JEJ' from this we get Z.ASJ = heliocentric longitude. Then retrograde longitude of N i.e. Z.ASN is added. We get Z.ASN + L ASJ = ^NSJ . From this heliocentric sara (latitude) is first calculated as in above case by multipliying with (MR and then geocentric valkue is obtained by R/K. ^NSP or ZNSJ has been called viksepa kendra i.e. heliocentric distance between pata and planet in both cases. Verses 12-26 : Further correction for sara - The above sara has been written according to old siddhanta which is inaccurate according to author. Now accurate sara of mangala etc as actually seen in explained. Sun and moon are to be corrected for parallax, when away from midday-sun (i.e. zenith), due to difference of observation from earth's centre and surface. Similarly, correction in sara is to make it sphuta (from heliocentric to geocentric position). Mean positions of mangala, guru and sani are substracted from their sphuta mandocca to get the manda kendra. Jya of manda kendra is mandaphala Conjunction of Planets 623 approximately. By adding or substracting this from mean position we get manda sphuta graha. Sighrocca of budha and sukra is substracted from their mandocca sphuta. For budha, its gighrocca is corrected by its parocca kandraphala. Result is sighra kendra for viksepa purpose. For viksepa kendra of other three planets, manda spasta graha is substracted from its pata. These are sara kendra of all 5 planets. From its bhuja jya, sara is found by multiplying with parama sara and dividing with trijya - heliocentric value. Sara is in north or south direction as explained in case of moon. Difference of third mandakarna and trijya is multiplied by difference of fourth sighra karna and trijya and divided by trijya. We get ksepa karnantara. Sara Karna - (1) When fourth sighra karna is more than trijya - (a) when third manda karna also is bigger than trijya - Karnantara is substracted from trijya (b) when third manda karna is less than trijya - ksepa karnantara is added to trijya. (2) When fourth sighra karna is less than trijya- (a) manda karna is more - then karnantara is added to trijya (b) when mandakarna is less - then karnantara is substracted from trijya. For budha and sukra, karnantara is added or substracted from mandakarna instead of trijya. Thus we get sara karna of all the five planets for all situations. 624 Siddhdnta Darpana Sphuta sara : As in previous method, pata is substracted from graha. Jya of this viksepa kendra is multiplied by madhyama sara and divided by sara karna. Quotient is multiplied by trijya and divided by fourth sighra karna to obtain sphuta sara of planets in kali. Its difference with sthula (rough) sara also can be used. Notes : (1) First we calculate the heliocentric position by mandasphuta graha as explained in spastadhikara. (2) Sara karna is real distance of planet from sun due to sara in its sighra gati. Difference of manda karna and trijya is proportional change of distance due to mandaphala. It is multiplied by proportional change due to sighra phala by multiplying with (sighra karna trijya) and dividing by trijya. When mandakarna or sighra karna is greater than trijya, sara karna i.e. true position of planet with sara, is less because sara will look smaller from larger distance. Hence sara karnantara is substracted from trijya, average distance. For budha and sukra average distance is their manda karna i.e. distance of sun from earth. « (3) Madhyama sara is value of sara seen from sun, it is multiplied by sara karna to get its true value as seen from sun. For proportionate reduction for geocentric value; it is multiplied by R/K. as explained in notes after pervious verse. Verses 27-31 : Ayana drk-karma : Sayana graha is added with 3 rasi (90 ° ) - which is satribha sayana sphuta graha. Its kranti Conjunction of Planets $ 2 5 jya is multiplied by sphuta sara and divided by dyujya of satribha sayana graha. The result will be in lipta etc. and is called ayana drkkarma kala. When ayana and sara of graha are in different direction, ayana is added to graha; and substracted if they are in same direction. Then graha postition or equator will be found, i.e. kadambaprota graha will become dhruva prota. This is called ayana drkkarma. After doing ayana drkkarma, again the dif- ference of planets involved in war (conjunction) is found. As before; the time is calculated when their rasi, kala etc. are equal. This will give lapsed or remaining days of conjunction. At the time of this conjunction, the planets are equal upto kalas. Then again sara is found; ayana drkkarma for new position will be done. By repeating the process, we get accurate time of equatorial conjunction when kala of the two planets are equal. Notes : In figure 6, EMQ and CMD are nadl ?mandala and kranti mandala respectively. P is dhruva, K kadamba and G the planet or grahabimha. PGA is dhruvaprota and KGB. kadambaprota. Then B is sphutagraha or position of the planet on kranti mandala. A ;ls called krta ayana drkkarmaE jgraha - i.e. point on ecliptic ^corresponding to equator pposition. MA may be called C jpolar longitude of the planet U modern terms. GB is Fjgure 6 . ^^ valana $26 Siddhdnta Darpana viksepa of G, which is almost equal to GA. From GAB considered plane triangle AB = M GXGA JyaB But, in GKP, LGKP = 90° + say ana graha = satribha graha, PK is measure of obliquity of ecliptic or parama kranti. x - ^ JyaGKP x JyaPK * a G = fi^PG Jya (satribha graha) x Parama Kranti Jya Dyujya Krantijya (satribha graha) x Trijya Dyujya Jya B = Trijya, as B = 90° in (1) Hence from (1) and (2) GA X Kranti jya of satribha graha Dyujya AB = ayana drkkarma, i.e. shift in position of planet on ecliptic due to inclination of axis and sara. Verse 31-37 : Aksa drkkarma - Square of ayana drkkarma in kala and square of sara are added. Square root of sum is the suksma sara. When suksma sara and kranti are in same direction they are added; otherwise difference is taken for sphuta kranti of the planet. This will be distance from planet to the equator on polar circle. Sun is always on kranti vrtta so its madhya kranti and sphuta kranti are same. By the method explained in Triprasnadhikara, for both the planets (in conjunction), from sphuta kranti, we find their cara, dinardha nata and Conjunction of Planets 627 unnata kala. Nata and unata kala separately multiplied by 5400 and divided by their half day give jya of nata and unnata kala respectively. Difference of cara asu of graha for madhyama and sphuta kranti is taken as kala and multiplied by jttta jya and divided by trijya. The result in kala is substracted from graha in forenoon (east half of sky) and added to graha in west half, if sara is north. For south sara, reverse process will be done. Then the graha will be corrected with aksa drkkarma. After that, difference of both graha is found and the time since conjunction or remaining till that is found. For conjunction time; again aksa drkkarma is done. After repeated procedure, both graha will be in same samaprota vrtta. Then, their north south difference in found on that circle. Notes : (1) Sphuta sara : Sara (or madhyama sara) is GB in figure 6 which is distance of the planet from ecliptic along the circle through kadamba K. Along this circle the distance of planet from equator is GB'. But distance from equator is calculated along . great circle through dhruva P. Hence the total kranti i.e. distance from equator is GA' = GA + AA'. We take as spasta graha, i not real planet G but its projection B on ecliptic. Hence, kranti of B is the real kranti. ?' ; First we have to calculate GA, which is given by GA = VqB 2 + AB 2 as ^GBA is 90° and AGBA is small and considered a plane triangle. f AA' is almost equal to BB' which is kranti of the sphuta planet i..e madhya kranti. 628 Siddhdnta Darpana Calculation of GA is really not necessary by the above formula, as we have already assumed GA = GB in derivation. (2) Bhaskaracarya has explained the drkkarma with difference in rising time on horizon due to sara of the planet. When the ecliptic position of the planet is rising on horizon, then due to sara, the real planet is above the horizon for north sara (down for south sara) and rises earlier (or later for south sara). The difference in rising time is known by drkkarma. One component of drkkarma depends upon ayana valana (i.e. inclination be- tween equator and ecliptic) and the other com- ponent depends on aksavalana (i.e. local aksamsa - inclination of local horizon or vertical with horizon or vertical of equator). These components are called ayana drkkarma and aksa drkarma. Figure 7 - Akla drkkarma In figure 7, NZSZ' is yamyottara vrtta of a place and NES is diameter of horizon in its plane. QQ' and AA' are diameters of equator and ahoratra vrtta. PP' is diameter of unmandala. EM is kranti jyi; AM is dyujya = R cos0 , where <p is aksamsa. Conjunction of Planets $29 Due to kranti, the planet rises earlier at position N, MN = kujya = R sin d tan0 where 6 is kranti. Its value on equator is ET where 1 kala = 1 asu in time. ET = carajya = R tand tan <p Due to sara, the planet at M on ecliptic is seen at K' in direction K, which is kadamba or pole of the ecliptic. (Sara is shown north, when K is north from P). LYMP = v = ayana valana. Thus due to sara, the longitude of planet is shifted by K'M' on diurnal circle, K'M' = s sin v where s is the sara. This is equivalent to shift of s Sin v I cos <p on equator, which is ayana drkkarma in asu. If we put R cos0 = Dyujya and R sin v = satribha kranti jya (approx), we get the formula for ayana drkkarma given earlier. Another component of sara MK' is MM' = s cos v, which is the sara in perpendicular direction to equator. Hence, sphuta sara is EM' = EM + MM' = R sin 6 + s cos v. Thus effectively the diurnal circle of ecliptic planet of M will be shifted to LM'L' parallel circle to equator passing through M'. Then the planet, will rise at position R' (corresponding to R on diurnal circle and T' on equator). Thus the rising time will be earlier by TT\ J XT' = ET' - ET = difference of carajya. This is the simplest and most accurate formula given in any siddhanta text. (3) Difference in carajya is difference at norizon, corresponding to half day length (dyujya <* radius of equator). At other times it is Proportional to natajya i.e. distance from yamyottara position A or L of the planet. Thus Aksa valana at ista kala _ Aksa drkkarma at rising Jya of nata kala ~ Half day $30 Siddhdnta Darpana Formula for aksa drkkarma at any other time has not been given by any other author. It is seen that aksa drkkarma is deducted from planet in east sky as it rises earlier for north sara. Since it will set later in west, half proportionate addition will be done. Verses 38-40 : Biihba of planets Five tara grahas like mangala have five types of birhba - madhya vrtta biihba, madhya bhasvara bimba, sphuta vrtta bimba, sphuta bhasvara birhba and drktulya birhba. Bimba of sun is very bright. Planets like moon take light from that and reflect it like water surface. Tara graha also have horns, due to the angle between direction of the graha and sun. But due to their distance from sun being large compared to moon, their horns are not seen. They are seen . as point only. Notes : (1) Birhba of tara grahas have been discussed in detail in birhbadhikara of siddhanta tattwa viveka, but this terminology has not been used any where. They are lighted by reflected light of sun, and bimba of sukra and budha are seen less than half when they are between earth and sun, due to their dark phase like moon. It has also discussed hole is sun due to sukra (like eclipse by moon). Due to smallness of taragrahas (small angular diameters) they are seen only as a point and their horns are not seen due to dark phase like moon. (2) From the context, the classification of bimba depends on their distance from earth, due to which they look small or big and due to phase Conjunction of Planets 631 i.e. dark part depending on angular distance from sun. Thus the classifications are - Distance difference - (i) Madhya vrtta birhba- Average bimba size seen at average distance. (ii) Sphuta vrtta bimba - Current size of bimba depending on the sphuta distance of planet. Phase difference (iii) Madhya bhasvara bimba - Half lighted phase corresponding to about 90° angular separation from sun. (iv) Sphuta bhasvara bimba - True lighted portion according to angular separation. Actual bimba - (v) Drktulya bimba -' which is actually seen according to distance and phase effects. Verses 41-42 : Diameters of planets Diameters of star planets in yojana are Mangala (450), Budha (930), Guru (4750), Sukra (2600) Sani (3500). These divided by 2213 give the bimba in kala in sun orbit. Notes : (1) Yojana value in sun's orbit is converted to kala by dividing it with 2213 as explained in candragrahana (chapter 8) verse 25. le made by 1 yojana at that distance is : — radian = ='- — kala sun's mean distance mean distance as 1 radian = 3438 kala hi, I' A & ■ ■ ■ . 3438 1 kala = tttt kala exact. mi:i m v 76,08,294 2213 This exact value indicates, that distance of sun has been calculated on basis of this ratio, after the diameter of sun was assumed 72,000 yojahas according to Atharvaveda. 632 Siddhanta Darpana All other text books have compared the diameters of planet in moon's orbit , but siddhanta darpana has compared them in sun's orbit. The linear diameter is based on accumption that the distances are inversely proportional to angular speed i.e. proportional to period of rotation. As the comparative distance of moon and sun on that basis was rejected due to correct looking value of Atharvaveda, reference to moon's orbit also was rejected. However, the distance of other planets and sun are considered proportional to their periods of rotation. This is justified because all planets move round the sun and moon around earth both according to siddhanta darpana and modern theory. Period of rotaiton T and distance D are not directly proportional, but according to Keplar's third law T 2 a D 3 where D is distance (semi major axis) Thus T a D 3/2 instead of Ta D assumed here. Thus, actual relative distance of farther planets will be lower than calculated here. There is evidence in vedas that orbit was not meant the linear circle, but the surface of sphere on which this circle moves due to rotation of pata. The same concept is used in Jain texts also. If time period is considered proportioanl to volume, then this relation T 2 a D 3 holds as T a 4/3 II r 3 , D 2 2 3 4ot where r is radius of orbit. Then T and D both ax . Time volume relation is only a conjecture Conjunction of Planets (2) Comparison of values Mean angular diameters of planets (1) 633 Planet Aryabhata I Vatesvara • Tycho Siddhanta Modern and Lalla Brahe Darpana (mean) Mars 1'15".6 1"19".2 1'40" 8" 14". 3 Mercury 2'6" 2'12" 2'0" 25" 9" Jupiter 3'9" 3' 18" . 2 # 45" 25" 41" Venus 6'18" 6'36" 3'15" 70" 39" Saturn 1'34" .5 1'39" 1'50" 10" 17" (2) Planet Old Surya Brahmagupta Surya Aryabhata Bhaskara II siddhanta and Sripati siddhanta & Bhattotpala n Mars 4' 4'46" 2' 4' 45" 4'45" Mercury T 6'14" 3' 6' 15" 6'15" Jupiter 8' 7'22" 3' 30" T 15" 7'20" Venus 9' 9' 4' 9* 9* Saturn 5' 5'24" 2'30" 5' 15" 5' 20" Aryabhata and Vatesvara have reduced the values of surya siddhanta and made them more correct. They are generally more correct then values of Tycho Brahe, who had observed with telescope. Old surya siddhanta value is 2 to 2-1/2 times the values of modern surya siddhanta and have been approximately followed by others in table (2). Siddhanta Darpana has evidendy reduced the value of angular diameters in ratio of about 11, in which ratio the diameter and distance of sun have been increased. However, compared to surya siddhanta ratios, he has made increase in mercury and venus diameters and reduced the ratio of outer planets. For outer planets ratio of siddhanta darpana and modern values are Mars 1 : 1.8, Jupiter 1/1.64 Saturn 1/1.7 Ratio for inner planets is Merucry 2.8/1, venus 1.8/1 634 Siddhanta Darpana One reason may be that, the visibility of outer planets reduces due to large distance from sun, hence they appear smaller. Minute values of old S.S/Seconds value of siddhanta Darpana for outer planets is Mars 1:2, Jupiter 1:3 Saturn 1:2 For inner planets Mercury 1:3.6, Venus 1:7.8 It is quite probable that Candrasekhara has calculated the angular diameters of inner planets according to their average distance from sun which is much less. Comparison of linear diameters : Planet Siddhanta Darpana yojana Earth=l 754.3 0.472 601.6 0.376 8324.5 5.203 802.1 0.501 14,776.4 9.235 Surya siddhanta relative figure are almost correct for all planets except Jupiter and venus whose value is about half the true value. These errors might have come due to incorrect ratio of time period and distance. However, sun's diameter comes to be less than, jupiter and saturn also as it is taken only 6500 yojanas. Figure of siddhanta darpana are more correct with two errors - Jupiter and saturn values reduced by about l/4th of correct value, mars about half value. But mercury and venus vahes have been increased about 1.45 and 1.63 times the correct value. This appears to be due to error in estimating angular diameters of inner planets (more). yojana Earth * 1 Mars 450 0.281 Mercury 930 0.581 Jupiter 4750 2.969 Venus 2600 1.625 Saturn 3500 2.188 [odemdiam Distance Earth = 1 Earth = 1 0.536 1.523688 0.403 0.387099 10.925 5.202803 0.990 0.723331 9.01 9.538843 Conjunction of Planets 635 Other correct feature is that all planets are assumed much smaller than sun (72,000 yojana diameter) due to more correct diameter of sun. Verses 43-44 : Madhya bimba The angular diameters of budha and sukra in sun orbit are their mean diameter. Angular diameters of other planets are obtained by multi- plying their angular bimba in sun orbit by sighra paridhi of the planet and dividing by 360°. The angular diameters in vikala are Mangala 8, budha 25, guru (25), Sukra 70 and sani 10 Notes : Average distance of budha and sukra is same as average distance of sun from earth as these inner planets are in small orbit round sun. For outer planets Distance of sun _ Sighra paridhi of planet Distance of planet 360° as sun is the sighra kendra for outer planets. Hence the formula for angular diameters. Verses 45-50 : Bhasvara and sphuta bimba - Madhyama bimba (angular diameters) are kept at two places. At one place, it is multiplied by utkramajya in the process of fourth sighra phala and divided by two times trijya (6876) Result will be substracted from the madhyama bimba at other place. This will be bhasvara bimba of the planet. For budha and sukra, when their sighra is in 6 rasis beginning from makara etc, then fourth phala kala and sighra koti kala are added; jya of the sum is added to trijya. Then 3 rasis are added to the sighra kendra of budha and sukra and its 636 Siddhdnta Darpana koti rasi is substracted. Bhujajya of the remainder is multiplied by bimba diameter and divided by two times the trijya. Result will be bhasvara bimba of these two planets. If sighra phala kala and koti kala is more than 3 rasi together, then jya of the sum is added to trijya to find multipliers for budha and sukra. Thus we get madhya vrtta bimba and madhya bhasvara bimba. They are separately multiplied by trijya and divided by their sighra karna to get the sphuta bimba and sphuta bhasvara bimba. Notes : (1) Surya siddhanta has given the following formula; spasta bimba Madhyama bimba X 2 x T rijya Trijya + Fourth sighra karna This formula is correct if the madhyama bimba is calculated at distance of sun, but in surya siddhanta it is calculated at the distance of moon. However, this formula is correct for siddhanta daprana where madhya bimba has been calculated at sun's distance. This is the second formula given here and is based on the following ratio. Trijya + 4th s ighra karna — : Trijya = Madhya bimba : spasta bimba. More accurately this should be found from true distance of planet from earth i.e. 4th sighra karna instead of average of trijya and sighra karna. Thus siddhanta darpana gives correct formula for spasta bimba based on ratio — Sighra Karna : Trijya = Madhya bimba : spasta bimba Conjunction of Planets 637 (2) Earlier correction : Bhasvara birhba is measure of relative visibility. If depends upon distance from sun due to which brightness decreases in ratio of square of distance (inverse square law). However, due to phase also the brightness increases and is more when angular distance between planet and sun is 90° to 270 °, we calculate utkramajya which deducted from trijya gives kotijya. Phase of a planet is equal to illuminated area divided by whole area of disc. The crecent GCHFG, bounded on one side by the semi ellipse GFH and on other side by semi circle GCH, is the illuminated part. GH is the line of cusps and CD the diameter perpendicular to it. Let CD = 2a. Then the phase Illuminated arc Area of disc G j-, ■: H ■ ■ E ^ To earth Figure Ba - Phase of the graha Figure 8b - Lighted part dueto sun £■ - n a (CM - FM) CF (1) n a 2 2a Now in figure 8b, hemisphere ACB is lighted sun and hemesphere CAD is seen from earth. £SME = d Hence CF = CM - FM = a - a cos AMC 638 Siddhanta Darpana = a (1 + cos EMS) = a (1+cos d) Hence phase = (1 + cos d)/2 (2) If we measure the difference between planet and sun, as d', then d' ■= 180* - d Hence phase = (1-cos d')/2 - - - (3) Thus in formula we find the utkrama jya R (1 cos d') and divide it by 2 R to get the phase according to eqn. (3). By substracting this portion from total bimba, we get the unlighted portion which is away from sun. For budha and sukra, the phase is calculated when they are on farther side of sun (sighra kendra 270° to 90°) when they are more illuminated. We approximately find distance of mercury from superior conjunction (adding 3 rasi to 270 is ) or inferior conjunction. One gives illuminted figure but on farther side, the other gives dark portion but on nearer side. Verse 51 : When sighra kendra of budha and sukra is 6 rasi i.e. they are between earth and sun then they are like black holes compared to bright sun in its disc. Verses 52-55 : Now observed bimba of bhasvara is stated. Bhasvara bimba appear sthula (i e round without sharp cusps) like a candle flame at far distance (which appears a round point instead of elongated figure) When a bright object is very far, it appears 215 times its real angular diameter. Bhasvara bimba kala is multiplied by 16 and square root of tne product is taken. That is observed value of seen bimba. m? Conjunction of Planets 639 Notes : (1) Reasons of this arbitrary assump- tion are not known. However, from the discussions three variations in bimba emerge - Sphuta bimba is linear change in angular diameter which decreases with distance - like diameter of moon and sun. Bhasvara bimba is the lighted portion of disc due to its phases like moon. Observed bimba of a point like object is seen 215 times bigger. But square root of bhasvara bimba is divided by 4 only for the diameter of observed bimba in kala. (2) Logic of this method is not under stood. A point like object will appear bigger due to diffraction or scattering of light. That increase in angular width will be fixed and not 215 times the radius. Its angular increase will be same for sun and moon also. Possibly Candrasekhar had seen some star planets with a telescope set at 215 times magnification as mentioned by Prof. J.C. Ray in his introduction. Verse 55 : Naksatras are self illuminated and their distance is fixed, as it is almost infinite compared to planetary distances. Still their seen angular diameter should be found out. Note : Though the stars are point like, two stars or star and a star planet are seen together, even when they are slightly separated. There are two reasons for that - Due to scattering of light in atmosphere, the point object appears to have, some width. Even when they are separated, their distance cannot be seen if it is less than limit of resolution Pf human eye. 640 Siddhanta Darpana Verses 56-60 : Types of conjunction - Now types of conjunction (yuddha or samagama) are being stated. (1) When the observed birhba of two planets touch each other, that is called ullekha yuddha (touching conjunction). (2) When birhba of • a planet enters another planet, it is called vedha or bheda yuddha (piercing conjunction). (3) When north south difference of two planets in conjunction is less then sum of semi diamters, then it is arhsa vimarda yuddha (part eclipse conjunction). (4) When the mutual distance is more than sum of semi-diameters, then it is called apasavya (i.e. separated), Then the difference is upto 1° (60 kala) i.e. Distance between centres - sum of semi diameters < 60 Kala. (5) When the separation is more than 60 kala then it is called samagama. (6) When, in an apasavya (separation less than 60 kala), one planet is bright and the other is dark (inferior planet between earth and sun), then it is called yuddha. When both are bright, it is called samagama When both are dark, it is called kuta yuddha. (7) When two planets are equal in longitude (i.e. in yuddha) and» northern planet has bigger diameter, then the southern planet is conquered. When both are equal, then north birhba is conquered, south is victor. Conjunction of Planets 641 Sukra is victor, whether in north or south (as it has largest birhba among tara grahas and is brightest). Notes : These are only conventions for predicting ftirture events and described in Brhat sarhhita etc. Here samagama has been used twice. One is conjunction when rim distance is more than 60 kala. Another is yuddha in which both planets are equally bright. However, conjunction of moon with a star has been called samagama generally. Verses 61-63 - South north distance To know the north south distance, two drkkarmas have already been described. As in eclipse, nata and larhbana corrections also are needed for the true north south distance. Earlier astronomers didn't observe or calculate less than 1/2 degree or 30 kala, hence they ignored nata and lambana of tara graha which is much smaller. Still for academic interest it is being described to explain the mathematics. < Verses 64-67 : Nati of if ptaW£ Parama nati of sun is 22 vikala. Madhyama nati of budha and sukra als6 in same. Nati Kala of budha and sukra (22/60) is multiplied by trijya and divided by last slghra karna. Quotient is again multiplied by vitribha natamsa (drkksepa) and divided by trijya for spasta nati of budha and sukra. For other three planets (mangala, guru and sani, parama nati of ravi is multiplied by their | iighra paridhi and divided by 360. Quotient is niultiplied by trijya and divided by fourth slghra $42 Siddhdnta Darpana karna. Result is again multiplied by drkksepa and divided by trijya to get the spasta nati. As in solar eclipse, viksepa of the 5 planets is corrected with spasta nati to get the sphuta sara. Notes : (1) Average distance of budha and sukra is same as that of sun, hence their parama madhya nati will be same as that of sun. As the parallax reduces in proportion to distance similarly for outer planets - me an parallax of planet _ mean distance of sun mean parallax of sun " mean distance of planet sighra paridhi 360° as sun is considered sighra kendra of outer planets. True parama nati mean distance (2} " = " ■ . Nfean parama nati True distance Trijya Fourth sighra karna (3) Parama nati is for horizontal position for which drkksepa or jya of vertical distance (south) is maximum or equal to trijya (R). Since nati depends on jya of vertical distance towards south spasta nati parama nati (4) Correction of sara for nati has already been explained for solar eclipse-. They are added if in same direction and subtracted if in different direction. Conjunction of Planets 643 Verses 68-71 : Laihbana correction At the time of conjunction, parama nati of the planet is multiplied by drggati and divided by trijya, and quotient is multiplied by jya of difference between planet with vitribha lagna and divided by trijya. Then we get sphuta laihbana (parallax in east west direction). When planet is east of vitribha lagna, sphuta laihbana is added to planet, otherwise substracted. After laihbana correction, some difference comes in the longitudes of the planets. Then again conjunction time is corrected when the longitudes are same. For this new conjunction time, again laihbana is calculated and, new conjunction time is found, when they will be equal in longitude. 1 After repeated processes, we get the true conjunc- tion time. Notes : Parama nati of the planet is found as above section. Drgjya is the distance of planet from * vertical direction and nati will be proportional to it. Its value in ecliptic is proportionately known from distance of planet from vitribha. This has been explained in solar eclipse- Verses 72 : Conmjunction of graha and naksatra - | Since naksatras are very far from earth, their | speed and parallax both are zero. Hence, its | conjunction with a planet is calculated only from | ttie speed of graha. i 3feises 73-75 : Bheda yuddha ' " Since lambana and nati are very difficult, this I correction is done only for rinding bheda yuddha, 644 Siddhanta Darpana when bimba of one planet enters the bimba of another. For other conjunctions this is not neces- sary. Bheda of sun by budha or sukra should be calculated like other conjunctions. When they are moving in opposite direction (budha or sukra is vakri), then from sum of the gati and when both are margl, by difference of gati, we calculate the conjunction. According to the respective sizes of bimba, times of sparsa etc can be found. Sara of vakri budha or sukra is very little so vedha of sun is done by them. In this case time of sparsa etc is found from sum of speeds. Verses 76 : Moon and star planets - Moon is corrected for nati and lambana and its vedha by graha bimba is calculated like sun. Verses 77-90 - Samagama of moon and star planets- When a tara graha and candra have equal longitude (rasi, amsa and kala), then for finding their lambana, madhya gati of moon (790/35) is divided by 14. Quotient (56/28) is reduced by lambana of tara graha found from its parama nati. This will be maximum value of nati difference of moon and that planet. Parama nati difference is kept at two places. It is multiplied by 60 (to make it vikala) and divided by madhyama gati difference of moon and the planets. If the planet is vakri, then it is divided by sum of gati. This is time of parama lambana in ghati etc; It is multiplied by drggati of that time (vitribha sanku) and divided by trijya (3438). Result is made asu. It is assumed kala and its jya is called 'para'. Conjunction of Planets 645 Bhuja and koti jya of difference between moon and lagna is found. Difference of bhuja jya and para is squared and added to square of koti jya. Square root of the sum will be chaya karna. Kotijya is multiplied by para and divided by chaya karna. Result will be madhyama lambana. Madhyama lambana is multiplied by difference of madhyama gati and divided by difference of sphuta gati if the tara graha is margi. If tara graha is vakri, then madhya lambana is multiplied by sum of madhya gati and divided by sum of sphuta gati. Result is spasta lambana. This lambana is substracted from moon, if it is east (more) of vitribha lagna, otherwise added. Then the new time of conjunction is found when moon and graha have the same lipta. The lambana asu is multiplied by second vitribha sanku and divided by 1st vitribha sanku (before lambana correction). After correction of moon by this sphuta lambana asu, we find the sphuta madhya kala of conjunction. According to method of solar eclipse, drkksepa of vitribha lagna at mid conjunction time is found. Its 1/513 is added and divided by 61 to find nati of moon. By method of solar eclipce, from nata jya of vitribha lagna, sara and aksarhsa valana are found. When sara of moon and graha are in same direction, difference is taken, when they are in different direction they are added. This sara will be useful for diagram (parilekha) of samagama. When graha is south from moon, sara will be yamya, when it is north, sara will be saumya. 646 Siddhanta Darpana For tara graha, moon is chadaka (eclipser) because it is closest to earth. Since moon has more speed, sparsa of its bimba by the planet will be in east and moksa will be in west. After doing ayana drkkarma of graha, graha and naksatra conjunction is calculated from nati corrected sara. Notes : The methods are exactly similar to methods of solar eclipse. Only difference is that the tara graha can be vakri also, when sum of gati is used instead of their difference. Verses 91-96 - Parilekha Like diagram of eclipse, we draw the manaikya vrtta (circle with radius as sum of semi diameters) inside khagola vrtta with radius 57 718' angula = 3438' radius. From same centre moon circle is drawn. For valana of khavrtta, sparsika valana in east and mauksika valana in west is given in their own directions. From valanagra, we draw a line to the centre of moon, called diksutra. From the points where diksutra cuts manaikya, we give sara at the time of sparsa and moksa in their direciton (north or south). The line from saragra points (end points of sara) to centre of moon, cuts the moon bimba on two points indicating entry and exit points of graha or naksatra. In conjunction of naksatra and moon, sanku of vitribha lagna is multiplied by 100 and divi<fcd by 231 to give jya of parama lambana or 'para'. Like moon and star/planet conjunction, vakri budha and sukra enter the sun disc from east side Conjunction of Planets 647 and exit from west side. Since sun has no sara, the sara of only budha or sukra is the total sara and direction of this will be the direction of sara. pise of sun will be in centre of samasa vrtta (circle with radius as sum of semi diameters). Notes : The discription in parilekha, chapter 10 is sufficient to understand this. Verses 97-106 : Observing shadow of planets From rays of star planets like mangala, we cannot see the shadow of a 12 angula sanku. Ilience, a mirror is kept on the shadow end point |ind sanku top is seen in mirror. Exactly at shadow pnd point, the planet and sanku end are seen in lone direction. On a plane level surface, we keep a vertical lanku of 5 hands hight. In it 12 divisions are marked, each being 1 angula. Sanku will be strong md straight and its surface will be cylinderical. As explained in Triprasnadhikara, from the iiata kala of the planet at desired time, we find the shadow length of 12 angula sanku. With that semi diameter a circle is drawn with sanku centre m the centre. Direction points are marked (earlier m day time) and from the centre, lines are drawn m east west and north south direction. Then the kranti jya of graha at the desired time is multiplied by chaya karna and divided by lambajya. Quotient will be karna vrttagra in angula. It will be substracted from palabha for north kranti "Si the planet and added for south kranti to get ija of shadow in angula (its distance in north south direction from sanku). On north south through centre, we mark a point at distance 648 Siddhdnta Darpana of chayagra bhuja in the opposite direction of inclination of the planet. From shadow length (chaya) square, we substract the square of chaya bhuja and take the square root. Result is dharatala sanku which is called koti also. When planet is in west half of sky, koti is given east from the end point of chaya bhuja. At the point of shadow circle where it cuts, shadow end will lie. At this point a tube will be kept in direction of the sanku top and we see from below. Or a mirror is kept and its reflection is seen. N W jr N 1 ^S p f N 2 * 1 I E W Figure 9 (a) Figure 9 (b) Notes - In figure (a) ENWS is circle with radius of shadow length. Current direction of shadow is OP. OP is length of shaodw, ONi is chaya bhuja, NiP is its koti. Hence OP 2 = ONi 2 + NiP 2 . ON2' is palabha, i.e. shaodw at the time of equinox midday. The difference with bhuja is karna vrttagra, N1N2' = ONi - ON 2 N1N2 - 7 Sin d' where 6' = spasta kranti, cos <p r . . / <p = aksamsa. This is explained in Triprasnadhikara Conjunction of Planets 649 In figure (b) OC is sanku of 12 angula length. OP is chaya and PC is chaya karna. Thus PC is in direction of planet at G. If we keep a tube in PC direction, planet can be seen from P end. By a long dark tube we can see a planet in day time also as scattered day light is absorbed by inner surface of tube and only light of planet is seen which is not obstructed. Alternatively, by keeping a horizontal mirror at P we can see planet by keeping eye in direction of PK, Here PK makes same angle with vertical PC as LCPC = d 1 in opposite direction. Verses 107-108 : Seeing the yuti At the time of yuti (conjunction of planets) we keep two sanku at the distance of sara difference and from the same point P we can see both planets through tube or a mirror. Result of different types of yuti are given in books of samhita (like Br hat samhita of Varahamihira). Verse 109 : Increased size of vrtta bimba - Here, the bimba of planets decribed or birhbas of stars to be told later, are very bright, hence they are seen 16 times more lighted than moon. At the time of sunrise and sunset, their discs are as bright as moon, hence their bimba value has been stated as 4 times = vT6 larger. Thus the real t angular diameter is 1/4 of the seen diameter. Notes : (1) This explains the logic of formula ! for observed bimba in verse 54. But it is not correct. (2) Due to diffraction of light, two points at angle less than 6 radians cannot be seen separately where 650 Siddhdnta Darpana Sin e ' ~5~ where D is diameter of aperture through which planet is seen (it may be aperture of pupil of eye or lens of a telescope). A is wave length of light (4000 to 8000 angstrom = 10* cm units). This is Raleigh criterion. Thus for visible light, when pupil is 1.5 mm diameter in day time, we cannot see two points which are separated by les than about 1' kala. In night time when pupil is bigger it will be about 20" vikala. Thus the angular diameters of outer planets are smaller than the limit of resolution of eye and even when they are separated, they appear together. This explains as to why separation upto 1 kala is called samagama and only for larger separation, they are really seen separate. Thus at the time of conjunction, the effective diameters of planets are seen digger. ' (3) Other reasons of fluctuation are scattering of light, and fluctuations in atmosphere, which are almost same for both the nearby stars or planets. The stars are so distant, that their angular diameter is zero even after seeing through largest telescopes. Their diameter of conjunction time is seen much more than 215 times due to diffraction. Verse 110 : Solar eclipse due to sukra To find eclipse of sun due to venus, their bimba and size of other tara graha is stated. In kali year 4975 (1874 AD) there was a solar eclipse due to sukra in vrscika rasi (i.e. in Nov.- Dec. month). Then sukra bimba was seen as 1/32 of solar bimba which is equal to 650 yojana. Thus it r Conjunction of Planets 651 |5 well proved that bimba of sukra and planets is much smaller then sun. Verses 111-112 : Prayer and conclusion May Lord Jagannatha remove our ignorance, who defeats beauty of blue clouds by his blue light aiid lives on sea coast. Thus ends the eleventh chapter describing conjunction of planets in Siddhanta Darpana written for tallying calculation and observation and education of students by Sri Candrasekhara, born |ii famous royal family of Orissa. &■■■ Chapter - 12 CONJUNCTION WITH STARS Verse 1 - Scope - To know the conjunction of planets with naksatras, the longitude and latitude of identifying star in each naksatra starting with asvinl, shape of naksatras and number of stars in it and bimba of yogatara (identifying star) is stated first. Verses 2-11 : Longitudes and latitudes of identifying stars (yogtara) S.No. Name Beginning Name of Longitude Latitude Position of of point yogatara of of of Naksatra naksatra longitude yogatara yogatara yoga tar a 1. Asvinl , 0' /?Arietis 10*07' +8*29' 10*07' 2. Bharani 13 "20* Arietis 24*21 +1027 11*01 3. Krttika • 26'4(T rfTauri 3608 +403 928 4. Rohini • 40*0' «Tauri 4556 -528 556 * 5. Mrgasira 53*2(y A Ononis 4951 -1323 631 6. Ardra 66*40* erOrionis 6454 -1602 -146 7. Punarvasu 80 '0' /JGeminorum 8922 +641 922 8. Pusya 93*20' <5Cancri 10452 +005 1132 9. Aslesa 106 # 40' crCancri 10947 -505 307 10. Magna 120*00' aLeonis 12558 +028 558 11. Purva Phalguni 133*20' £Leonis 13727 +1420 407 12. Uttara Phalguni 146*40' /JLeonis 14746 +1216 106 13. Hasta 160*0' <JCorvi 16936 -1212 936 14. Citra 173*20' aVirginis 17959 -203 639 15. Svati 186*4(r aBootis 18023 +3046 -617 16. Visakha 200*0' aLibra 20114 +020 113 17. Anuradha 213*20' <$Scorpii 21843 -159 523 18. Jyestha 226*4C aScorpii 22554 -434 -046 19. Mula 240*0' AScorpii 24044 -1347 044 20. Purva Asadha 253*20' dSagittarii 25043 -628 -237 Conjunction with Stars 653 2i. Uttara Asadha 22. Sravana 23. Dhanistha 24. Satabhisaj 25. Purva bhadrapada 26. Uttara bhadrapada 27. Revafi 266* 4C oSagittarii 25832 -327 •0' • in' 280*0 293*20 306 '4C 320*0' 333*20' 346*40' aAquilae ^Delphini AAquarii aPegasi yPegasi £Piscium 27755 29229 31743 32938 34518 35601 +2918 +3155 -023 +1924 +1236 -013 -808 -205 -051 1103 938 1158 921 These are the modern positions and names of identifying stars. Nirayana longitude of Citra (a-Virginis) was fixed as 180° at 285 AD to fix the nirayana position accurately in zero ayanamsa year. Now it has become 179° 59' due to negative proper motion of citra. Verse 12-24 : Verses 12-14 give the number ' of stars in each naksatra. Verses 15-18 give the shape of each naksatra. Verses 19-22 give the direction of yogatara within the naksatra (this can be known from their latitude and position in naksatra also given in previous table). Verses 23-24 give the diameter of yoga tar a in vikala. Actually the diameters are almost zero even by talescope viewing, they are measures of visual magnitudes of brightness. The yogatara positions of 28 naksatras including Abhijit according to siddhanta darpana in previous verses and the other details are given in chart form. SI. Naksatra No. 1 Asvini 2 Bharani 3 Krttika Owner (yajurveda) Asvina Yama Agni Yogatara Latitude Bimba Shape No. Longitude Vikala of Stars 9*45' 2100 3515 +10*30' +1100 +415 2 3 Horse 3 mouth Triangle 3 Flame 6 654 Siddhdnta Darpana 4 Rohini • Prajapati 4630 -537 7 Cart (Sakata) 5 5 Mrgasira Soma 6015 -1330 2 Cafs paw or head of dear 3 6 Ardra Rudra 6500 -1540 7 Coral or water drop 1 7 Punarvasu Aditi 9015 +630 8 Bow 5 8 Pusya Brhaspati 10400 +115 2 Arrow 3 9 Aslesa ■ Sarpa 10800 -1200 4 Dog tail 5 10 Magha Pitr * 12600 +022 6 Plough 5 11 Purva Phalguni Bhaga 14300 +1200 12 weight on two ends of beam 2 12 Uttara Phalguni Aryama 15300 +1300 13 -do- • 2 13 Hasta Savitr 16500 -1100 4 Hand 5 14 Citra Tvasta ■ ■ 17900 -210 7 Pearl 1 15 Svati Viyu 19300 +3300 13 Coral orjewel 1 16 VisSkha Indragni 20700 -200 2 Shed or tent 5 17 Anuradhd Mitra 21830 -200 4 Snake hood 7 18 JyesthS Indra 22530 -415 7 Teeth of Boer 3 19 Mula Nirrti * 24040 -1330 5 Cronch or lion's tail 9 20 Purva Apah 25000 -630 4 tusk 4 * Asadha • * 21 Uttara Asadha Vi§vcdavah 25630 -340 4 Chute (Supa) 4 Abhijit Brahma 25630 +6200 14 Triangle or fire bail 3 Conjunction with Stars 655 22 Sravana Visnu 27300 +3000 7 Arrow or short men 3 23 Dhanistha Vasava 28530 +3600 3 Long drum 5 24 Satabhisaj Varuna 31745 -020 3 Canopy 100 25 Purva bhadrapada Aja-Ekapada 32200 +3200 4 Cot or weights from beam 2 26 27. uttara bhadrapada Revati Ahirbudhnya Pusa * 33800 0*00' +2800 +500 4 3 do drum rtr fi«Vi 2 32 Notes : (1) Yoga tara in north position of naksatra - (1) Asvini 5. Mrgasira 11. PQrvaphalgunl, 16. Visakha 20 Purvasadha 21. Uttarasadha, 25 Purva Bhadrapada, 26 - Uttara bhadrapada Yogatara in centre - 19. Jyestha, 22 Sravana, 17 Anuradha, 3. Krttika, 8 Pusya Yogatara in isana (north east) - 7 Punarvasu, 13- Hasta, 19-Mula. Yogatara in west - 23. Dhanistha, - Abhijit Yogatara in east - 4. Rohini, 9. Aslesa Yogatara in south - 10. Magna (very bright), 27. Revati 12. Uttara phalguni Yogatara in agni kona (south east) - 24. Satabhisaj. Single stars are in 6. Ardra, 14. Citra and 15. Svati, hence there is no difference between the naksatra and yogatara. (2) Shape . of naksatras have been decribed dMferently by different authorities. Actually, it is only imagination and convention. (3) Longitudes and latitudes also differ slightly according to different authorities. 656 Siddhanta Darpana (4) It may be seen that many yagataraa do not come within extent of their naksatra. Hence three naksatras are divided into purva and uttara part. In unequal division of naksatras, most of the naksatras have yogatara in their extent. (5) It has already been stated that diffraction and partly scattering of light in atmlsphere spreads the point like stars. Bright star has bigger spread as, greater spread of diffraction ring remains visible. Verses 25-40 - Other stars - Now many other stars are described. (1) Lubdhaka (Sirius) - It is brightest star south of punarvasu with birhba of 20 vikala, dhruva 77° and dhruva prota kranti 40 ° . Surya siddhanta name of this star is lubdhaka. Bhaskara II has given its longitude (polar) as 86°. It is 8.6 light years away and brightest star. (2) Mrgavyadha - There is another small star south of punarvasu. Surya siddhanta and Lalla have called this same as lubdhaka, but it is different star. Its dhruva is 56°, south sara 32° and birhba is 10 vikala It may be identified with Orion, which is also called hunter is greek stories borrowed from Egypt. Its south latitude is same as south latitude of Magadaskara (now Malagasi) an island in south east direction of Africa - hence this island was called Mrga or Harina dvipa (3) flvala - This is a group of three stars between mrgavyadha and ardra. Its middle star is Conjunction with Stars 657 yogatara, whose dhruva is 61° and south sara 23°30'. (4) Hutabhuk - According to siirya siddhanta, its dhruva is 52° and north sara is 8\ (5) Brahmahrdaya - According to siirya siddhanta, its dhruva is 52° and north sara 30°. (6) Prajapati - It is 5° east brahmahrdaya whose dhruva is 57° and north sara is 38*. (Surya ,siddhanta) Modern observations have indicated the fol- lowing positions {by author). (4) Hutabhuk - Dhruva 58° 15', sara 5° 15' north birhba 6" vikala (5) Brahmahrdaya - Dhruva 56°, north sara 23°, birhba 16" (1) Lubdhaka is now called prajapati. (7) Apamvatsa - This is 5* north from citra. (8) Apa - This is 6* north of Apamvatsa. It is also called apyavasu. Dhruva of both (7) and (8) above are equal to citra. North sara of (7) is 2 - 50* and (8) is 8°5(r. (9) Agastya - Its dhruva is 95" and south sara is 75°. Its dhruva becomes sphuta after doing ayanamsa correction. Its bimba is 18* vikala. (10) Yama - Its dhruva is 22% sara is 66* south and birhba is 8". Surya siddhanta has stated dhruva of agastya as 90°. This was the value at the time of writing that book when 121 years were remaining in satya yuga. In Kali era 4251, Bhaskara II has stated its 658 Siddhdnta Darpana dhruva to be 87°. He has stated dhruva of punarvasu as 93° and Agastya 6° less i.e. 87°. At the time of siddhanta darpana, it is 17°30' west from punarvasu i.e. 90 "IS 7 - 17° 30' = 72° 45'. From agastya dhruva 95% on substracting ayanamsa 22 *, we get the same value 73° approximately. The change of agastya dhruva from 87° at the time of siddhanta siromani when ayanamsa was 11° 30' to mithun 13° (73 °) is the change in 719 years (1869 AD). ♦Notes : Ayanamsa correction is not needed when the distances have always been measured with respect to fixed stars. There may be some error in identification of stars. Otherwise relative motion of stars is very little and negligible compared to ayana movement. Opinions differ regarding correct identifications of these stars with current greek names used. Modern names of yogatara have already been given. Agastya is canopus, apamvatsa is 0-virginis and Apa - <5 vir- ginis, Agni or hutabhuk in p tauri. Prajapati is ft aurigae, Brahma is a aurigae. Verses 41-56 - Saptarsi mandala Since saptarsi mandala (great bear) is moving, its dhruva has not been stated by earlier astronomers. Still, I state their position, based on my experience. In north direction saptarsi mandala spread in east west direction like a bullock cart is very prominent in the sky. It has been most reverred in sarhhita and purana. Conjunction with Stars 659 Within this group, there is an upward raised line towards east. Marici is in its front. Behind it Vasistha is with Arundhati. Still west from Vasistha is Angira. After that, is a quadrilateral. In its isana kona (north east), lies Atri. South from it is Pulastya and west from Pulastya in Pulaha. North of Pulaha is Kratu. The great circle joining Pulaha and Kratu, cuts ecliptic in some point, the naksatra or rasi of that point is considered the rasi of saptarsi. At present Pulaha and Kratu are in 21° of sirhha i.e. 3rd quarter or purva-phalguni. 13 kalamsa east from them is Pulastya. Atri is 5 kalarhsa east from Pulaha, 9 kalamsa east from Atri is Angira, 8 kalamsa east from Angira lies Vasistha and 8 kalamsa east from Vasistha is Marici. Arundhati is a very small star, east from Vasistha which is barely visible and can be seen with telescope. This is not giver of good or bad omen, like the seven main stars. Its birhba is 1 vikala. Birhba of Atri is 3 vikala, and all others are 8 vikala. Mutual distance between these stars is same and equal to 10 pala kalam'sa. This 10 pala is multiplied by 1800 and divided by rising time of that rasi at equator. The quotient is added to the dhruva of Pulaha or Kratu (sirhha 21° = 141°). We get the dhruva in rasi etc for other stars. East west angular distance (along ecliptic) of saptarsi is 43°, but due to its position in sayana kanya and tula, it appears 46° (in rising time at equator). 660 Siddhania Darpana Distance from ecliptic along dhruva prota vrtta (great circle through dhruva, not kadamba ,- pole of ecliptic) in north direction are - Kratu 56°, Pulaha 51', Pulastya 53", Atri 59' Angira 60*, Va&stha 62° and Maria 60°. If sara of Vasisttha from kranti is fixed, then in the end of even quadrant, it wil be 4* from dhruva (North sara 62° + kranti at end of even quadrant 24° = 86' i.e. 4° from dhruva at 90°). Even if sphuta kranti of saptarsi remains same, their sara changes with change in rasi. P a u " x ^ - - 8 a m * s Polasis s v P' s Horizon N a, ;' S. *> North Pole a Aksamsa North Figure 1 Position of saptarsi and pole star , Notes (1) Due to earth's rotation, saptarsi makes a revolution around north pole in direction of line pa of its western stars. Three positions at 3 hour intervals are shown from east to west. Polaris P' is very close to north pole (58' Kala distance) and is called pole star. P is <p angle above north horizon, where is local north aksamsa. The stars are indicated by greek letters starting from Stars Greek nam 1 Marici rj Alkaid r 2 Vasistha £ Mizar 3 Angira e Alioth 4 Atri 6 Megrez 5 Pulastya y Phad 6 Fulaha p Merak 7 Kratu a Dubhe Conjunction with Stars 661 western lower star. Siddhanta counts them from eastern end. Modern names, distances and visual magnitudes are given below - Visual magnitude Distance in light years 1.87 210 2.06 88 1.79 68 3.3 2.44 90 2.37 78 1.81 107 More visual magnitude indicates lesser bright- ness, thus Atri is least bright and farthest/ hence its bimba vikala has been indicated small. Arundhati is a small star, below Vasistha called Alcor (magnitude 5). Mizar (Vasistha) itself is a double star when seen from telescope. It appears that Atri has faded now but earlier, it was equally bright. (2) Mythology : Callisto was attendent of goddess Juno but was more beautiful. To protect her from jealosy, Callisto was turned into bear by god Jupiter. When her son Areas, thought her a bear and wanted to kill her, he was also turned into bear (ursa minor) According to Puranas, Saptarsis are mental sons of Brahma. There is a separate set of sap tarsi for each of 14 manu periods of which 7 are yet to come. Rsi and Rksa have been used for star, sage or bear also. Hence ursa in Persian means saint, in Greek it means bear. Like this bear around north 662 Siddhanta Darpana pole, Russian bear exists. Russian was Rsika and it is land of bear. Proverbially Russia is called Russian bear. Rsi denoted sage and bear as both had long hairs. Hence the name great bear came. (3) Motion of Saptarsis : Only Vatesvara siddhanta chapter 1 verse 15 has given the number of revolutions of saptarsi which is 1692 in a yuga. On that basis, Karana sara of Vatesvara has given a method to calculate movement of saptarsis, as quoted by Albirun! (India I, page 392) - Multiply the basis (i.e. years elapsed since beginning of saka 821) by 47 and add 68000 to the product. Divide the sum by 10,000. Quotient is position of saptarasi in rasis etc. According to this formula, saptarsi has a motion of 47 signs per 10,000 years which is equivalent to 1692 revolution in 43,20,000 years, as stated above. The position in saka year 821 (Kali year 4000 was) 1692 x 12 x 4000 . A . . , 68,000 signs = 1 revolution + innnrt signs 43,20,000 * B 10,000 This accounts for the addition of 68000 in formula. (4) The stars of the constallation of the saptarsi do not have a motion relative to naksatras. So the statement of revolution is not correct. This appears to be the reason why many standard astromers like Aryabhata, Brahmagupta, Sripati, Bhaskaras I and II, Surya siddhanta etc do not deal with the subject at all, as being outside the pale of astronomy. Therefore, Kamalakara was constrained I Conjunction with Stars to say in his Siddhanta Tattva Viveka, Bhagraha yutyadhikara, verses 25-36 - ''Sage Sakalya has given the motion of the sages with their positions in his time. Surya and others who explain the nature of the celestial sphere in their works do not give it, and therefore, the theory cannot be sustained astronomically. Even today, this motion mentioned in the samhitas is not observed by astronomers. Therefore, the seven real sages who are the presiding deities (of these stars) are only to be supposed to be moving unobserved by men, for the prediction of the fruits, thereof." But the motion has been accepted as a fact by certain common people and authors of the Puranas, and an era called Laukika era by the peopie of Kashmir region and saptarsi era by the puranas have been founded on this theory. Mahabharata mentions, that when Yudhisthira ascended the throne, Saptarsis were in magha naksatra. Vayu purana chapter 99, tells that saptarsi's remain for hundred years in one naksatra. Hence they complete the round of 27 naksatras in 2700 divya years. However, same purana chapter 57, t" tells that saptarsi naksatra is of 3030 human years. Hence human year appears to be taken as 12 sideral revolution of moon. Divya year here means 1 solar year. 2700 solar years = 2700 X 365.256263 days for sidereal years 664 Siddhanta Darpana 2700 x 365.256263 , = 12 x 29.321661 lunar ^ sidereal = 3007.968 years Varaha Mihira has written in Brhatsamhita 13/3, that according to Vrddha Garga, Saptarsis were in Magna in the rule of Yudhisthira. Raja Tarangini of Kalahana has followed this era only in writing ancient history. (5) Explanation : Kamalaakara has explained that it has no relation with astronomy and it is only for astrological predictions. Siddhanta Darpana has tried to justify the movement of saptarsis on basis of their measurement of position on dhruva prota vrtta on ecliptic. Normally kadambaprota vrtta is used for ecliptic and dhruva prota for equator. This causes difference and has explained the difference in terms of ayanamsa. His calculation of difference from Bhaskaracarya time is based on ayana - movement. However, this will have a cycle of 26000 years and not of 2700 years of saptarsi era. Hence Candrasekhara has not mentioned the saptarsi era but has vaguely tried to justify its movement. My explaination is based on basis of vedanga jyotista which was current in Mahabharata period, Rk jyotisa has a cycle of 19 years in which 5 years are of samvatsara type (starting between Magha sukla 1 to Magha sukla 6.) Yajus jyotis starts with 5 year cycle of 366 days each, but this also becomes equivalent to 19 year cycle with 6 ksaya samvatsara in 5 cycles of 5 years each. P.V. Holey has assumed a bigger yuga of 19 X 8 + 8 = 160 years because (Conjunction with Stars 665 it gives very little error, but has not explained the mechanism of arranging the last eight years. This is not corroborated any internal evidence in the text. However, saptarsi era was very much in use and was accepted in the calendar system. This appears to be based on system of naming a century (100 solar years) on a naksatra in same way as we name every guru varsa or other solar years on basis of 1st week day of the civil year. Thus, we can name the century on basis of first naksatra of moon (or may be sun) in a century, if the vedic yuga system is followed. Then bigger yuga should be 19 X 5 + 5 years = 100 years instead of 160 = 19X8+8 years. Thus in a century, we can take last 5 years as the first 5 years of the 19 years Rk cycle Or 5 years first cycle of yajusa jyotisa. In taking yajusa cycle, the 19 year cycle doesn't break and in sixth cycle first 5 years are subcycle which make a century. It can also be seen that all cycles of 19 year start with Sravistha but after five years of yajus cycle, sixth year starts with satabhisaj which is the next star. Thus on completion of 100 years, |ln this calender, moon gains one naksatra in this jpfcalertder. Thus each sucessive century will start pWfch one naksatra later, which will be saptarsi jlaksatra or naksatra of the century. Sri Holey has opined that Rk jyotisa was itely written before 2884 B.C. According to traditions vedic texts or puranas were written 300 years of Mahabharata war. Thus it y indicates the calender system fixed in dhisthira time who had really started an era. !?,•• 666 Siddhanta Darpana That time year started with Magha sukla paksa, hence saptarsis were assumed to be in magha to start with. (6) Mutual distance between stars of saptarsi mandala is not equal as stated here. Verses 57-59 : Kranti of circumpolar stars Sphuta kranti of yama and Agastya is always same. Kranti of naksatras starting with Asvini keeps changing. Kranti of saptarsi mandala is fixed according to some, but changes according to others. Earlier astronomers have assumed motion of saptarsi mandala as 8 kala per year from east to west. But I (author) have not seen such gati. Hence I do not agree to it. Ayana sanskara of yama, Agastya and saptarsi mandala has been instructed to be done in opposite direction. This is valid for author's time only (1869 AD). Notes (1) Ayana samskara is needed because, here dhruvamsa have been given for yogataras and other stars. As stated earlier, Candrasekhara has assumed oscillatory motion of ayana, and according to this, the present backward movement will change after 2200 AD. (2) 8 Kala movement of saptarsi is 800 Kala or 1 naksatra in a century whcih has been stated by Vatesvara. This has not been accepted by author correctly. (3) Circumpolar stars are near dhruva (pole star) and appear to move round it. This is true for south polar region also. This depends on local latitude of the place. Day length of a planet or Conjunction with Stars 667 t r increases by carajya which is increase in half day length for north kranti. It is decrease for southern kranti. Increase in half day = Carajya = R tan d tan <p (<3 =spasta kranti, <p= latitude) If this is equal or greater than R, then the increase is equal to half day of equator itself and there will be no night i.e. the star will never set. For this tan <f> tan <5 > I Thus for any star with north kranti, it is circum polar, if tan 6 > cot <p Similarly for south kranti, also, if Tan d > cot <p then the star will never rise. Verses 60-62 : North pole star (dhruva tara) has birhba of 4 vikala. This is not the real position of dhruva i.e. pole of equator and dhruva prota is not drawn through it. The seen dhruva tara is 1°24' away from surface centre of equator. Hence, when revati naksatra comes on meridian i.e. dhruvatara at beginning of mesa, appears 84 Kala above the pole. When sravana, punarvasu naksatra are on yamyottara, dhruva rises above horizon equal to local aksamsa. Again, when citra naksatra comes on yamyottara, dhruva in 84 kala nata from its kendra. Notes : Polaris ( Ursae minoris) is a star of second magnitude and is 58' Kala away from celestial pole in west direction in 1950 AD. Celestial 668 Siddhdnta Darpana pole is moving towards polaris upto 2105 AD when it will be only 30' away, then will begin to recede from it. It may be seen that Draco or dragon group is pole of ecliptic i.e. pole of solar system. Thus Polasis 1900 AD a 8400 AD / 2700 BC 4600 BC Hercults a'Ly 14800 AD Figure 2 - centre of this circle is ecliptic pole proverbially sun as visnu is under draco or sesanaga with 3-1/2 turns. Base of human body cakras is also called serpent of 3-1/2 turns (Kundalini). Thus it was called draco is Chaldia and dragon of 3-1/2 turns in China also. Since pole star is 84 kala from pole to mesa Q , or revati naksatra, it appears above north pole when revati is on meridian and below 84 kala when naksatra 180° opposite citra is on meridian. When naksatras 90° from these are on meridian, (Sravana or punarvasu), altitude of north star will be same as pole (though east or west by 84 kala). Verse 63 - Similarly south pole star also appears to move around south pole like a bullock rotating the oilseed crusher in a circle. Conjunction with Stars 669 Note - There is no conspicous star near south pole. Octans group contains south pole, %ut its brightest star v is of 3.7 magnitude and official pole star a (sigma octanis) has 5.5 magnitude. It is in line with bigger arm of south crpSs group. VersesN>4-66 : Three measurements - Three angular measurements (west to east) are used - ManSrhsa, Kalamsa and Ksetramsa. Rising times of rasls being different, Ksetramsa and Kalamsa are different. After one revolution, both complete 360 \ Manamsa and Kalamsa are same on equator, but difference between them increases as we go further from equator, in north or south direction. '.' Notes : The different measures depend on different system of coordinates shown in figure 3. P = QE K = Figure 3 - System of Coordinates Celestial Pole (Dhruva) = Celestial equator* * Pole of equator (Kadamba) g 7 Siddhanta Darpana YL' = Plane of the ecliptic Y= First point of sayana mesa (vernal equinox) L,L' = First points of sayana makara and karka/winter and summer solstice). S = a heavenly body PS = A great circle through P,S, cutting equator at Q and ecliptic at B. Y Q = Right ascension = a = Kalamsa (time is measured along equator rotation) 1 Kala at equator = 1 asu, R.A. of 1 hour = 15° at equator. QS = Declination = Kranti 6 KS = Great circle through K, S, cutting ecliptic at C. Y C = Celestial longitude = A = Ksetramsa or bhogamsa CS - Celestial latitude = p = Sara or viksepa Y B = Polar longitude or dhruvaka = 1 BS = Polar latitude = viksepa (dhruva) = d Polar longitude (dhruvarhsa) and latitude (viksepa) have been used only in this chapter to indicate position of stars as we observe them with reference to fix position of pole. WE' is a circle parallel to equator in north at ange d, latitude of the circle Y. M = <f> where M is point on P corresponding to mesa 0°. Absolute length of arc between M and corresponding position Q' is almost same as great circle ^between them. The great circle between M and Q' lS manarhsa Arc MD = Arc Y Q. cos 6 Conjunction with Stars 671 Though the angular difference between MQ' and YQ is same, manamsa is less. It becomes lesser, if d increases i.e. we go farther from equator. Kalamsa is the distance along equator, hence it is equal to rising time of rasis (at equator). Verses 66-68 : Saptarsi measures Stars in saptarsi are taken from east to west along declining longitude (desantara), not in north south direction (aksamsa) (Second half of verse 66) Here difference between dhruva viksepa of pulaha and kratu is only 5°. Similarly, east west difference manamsa between kratu and marici is 25° and difference in viksepa is only 0°30\ Sphuta kranti can be calculated. Verses 69-70 : Conversion of three measures Sphuta kranti and dyujya are calculated. Sum of two dyujya for kranti vrtta and visuva vrtta is made half - it is called hara. Manamsa multiplied by trijya and divided by hara gives ksetramsa (degree on ecliptic) Ksetramsa multiplied by rising time of its rasi and divided by 180° gives kalamsa. This way kalamsa of marici and kratu can be found. By reverse process, manamsa can be found from this ^sa. Notes : (1) In notes of previous section, vide e (3) manamsa is measured along WE' parallel *p equator which is diurnal circle of a star of this on. It is easier to convert it to kalamsa as ined in the note, or in spastadhikara for Sacculation of day length. Manamsa M = H cos <5 672 Siddhdnta Darpana (where H is kalamsa, d is kranti) H. R cos d _ H Dyujya R Trijya However, d here is measured along dhruva prota SQ instead of SC line. Thus length along ecliptic is reduced due to lesser rising time of B compared to C, and increases due to oblique length of B. Thus instead of dyujya we take average dyujya and trijya. M Trijya Kalamsa H = -(dyujya + trijya) (2) Ksetramsa is converted to kalamsa as per the following approximate ratio used for calculation of lagna - Rising time of rasi Rising time for ksetramsa Rasi (1800 kala) " Kalamsa in kala Rising time of ksetramsa is measured along equator, hence its asu is equal to kala of kalamsa. Verses 71-75 : Variation of kalamsa and manamsa- Dhruva star moves in a circle of 360°, hence its manamsa 360 X 84 otAOtt = — = 8'48" 3438 i.e. 1° of this circle is equal to length of 8'48" on equator. Due to change in kranti, if shape of sapktarsis remains fixed, then with change in kranti, their rising time will also change with change in dyujya. Since kalantara of saptarsis is fixed with change in kranti then, east west distance will Conjunction with Stars 673 change with change in aksamsa. If kranti is fixed, then kalamsa will be constant. If ksetramsa is constant, then kalamsa and manamsa wil vary. Like kalamsa and bhagamsa, relation between kalamsa and manamsa also can be found. Verses 76-79 : Sara of naksatras The dhruvamsa of naksatras given here are already with ay ana drkkarma. Their sara also is sphuta i.e. in dhruva prota vrtta. But sara of graha is asphuta, i.e. in kadamba prota vrtta. After drkkarma, it will become sara in direction of dhruva prota vrtta of stars. Even when the. kadmba prota sara of naksatras is same, their kranti in dhruva prota circle will be different due to east west difference. Hence the length of their day and night will be different (as the semi diameter of diurnal circle . - dyujya, depends on distance from equator in dhruva prota direction). If the sphuta kranti of a naksatra is more than the co-latitude of a place, the naksatra will be always rising at that place (it will be always seen there above horizon). If the south sphuta kranti is more than the co4atitude of the place, it. will never rise above horizon, i.e. always set. Notes : Dhruvamsa has already been ex- plained. This has been used for indicating position of stars because it is easier to observe them with reference to north pole. Circumpolar stars have already been ex- plained. For them, carajya = R tan d tan <l> is 674 Siddhanta Darpana bigger than R, radius of equator, hence day length will be more than day night value. Thus tan d tan <p > 1 or tan 6 > cot <p = tan (90° - <p) Here, d is kranti, O is aksamsa, hence 90 ° -O is lambarhsa. Thus <5> 90° - O Then the star will be always rising if kranti is bigger than lambarhsa. Similarly for south kranti, if carajya is bigger than R, day length wil be less than 0, i.e. the star will not rise. This means the same condition. Physically, we can understand it, because north pole is above horizon at angle equal to local aksamsa. Distance from north pole to the star is 90° - d which should be always less thanO if the star is to remain above horizon. Thus <p> 90* - d or 6 > 90° - O i.e. Kranti > lambamsa Similarly, south pole is 0° below south horizon, A star with south kranti will be 90° - d away from south pole. If this distance 90° - d is less than O , then the star will never rise. * > Verses 80-84 - Conjunction of graha and naksatra Ayana drkkarma is done for the involved graha and difference of dhruvarhsa of graha and naksatra is found. The difference in kala is divided by sphuta gati of the graha in kala to get result in day, ghati etc. If dhruva of graha is less than naksatra, the conjunction will occur after that interval, if it is more, then the yoga has already Conjunction with Stars $75 occurred, that period before. When graha is vakri (retrograde) then opposite order will happen (i.e. if graha dhruva is less, conjunction has happened earlier). For this conjunction time, we again find difference in sphuta dhruva of graha and naksatra and get the more accurate value of conjunction time. After successive approximations, we get the correct conjunction time. After that, sphuta kranti and cara of graha and naksatra are found and cara is calculated. That ; will give periods of their day and night. From that | we get the values of udaya and asta lagna of graha I and naksatra for their rising and setting times. As \ explained before, the rising and setting times will j be when sphuta sun reaches those positions (of I udaya and asta lagna). By finding difference of arhsa at rising setting times, we get the proportional \ difference between graha and naksatra according !' to the natakala (aksa drkkarma explained earlier) and again we revise the conjunction time, when longitude of graha and naksatra are same after aksa drkkarma. * As explained in conjunction of planets, we I find the north south difference between graha and naksatra from difference of their dhruva prota sara. I Distance between discs is obtained by substracting I the semi diameter of bimbas from this distance. I Notes : The methods of ayana and aksa fdrkkarma have already been explained in conjunc- tion of planets. Here, the problem is simpler, because the position of naksatra is alrady stated corrected by ayana drkkarma. Further, we need *u>t calculate motion of naksatra, because they are gfixed. Here also conjunctions will be different Recording to distance between discs. $76 Siddhanta Darpana Verses 85-87 : Bheda of naksatras Planets can enter the following 13 naksatras (or do 'bheda' in their extent) - Rohinl, pusya, krttika, citra, magna, punar- vasu, anuradha, jyestha, visakha, revatl, satabhisaj, purvasadha, and uttarasadha. Other fifteen naksatras are never crossed by planets (no bheda) - Asvinl, bharani, mrgasira, ardra, aslesa, purva phalguni, uttaraphalgunl, hasta, svati, mula, ab- hijit, sravana, dhanistha, purva bhadra pada and uttara bhadrapada. Among crossed (bhedya) naksatras, punarvasu is crossed by every planet. Purvasadha, revatT, and krttika are sometimes crossed. Others are less frequently crossed according to kranti of the graha. The planet whose south kranti in 14th degree of vrsa (44°) is more than 2° 20', can cross the rohini (in shape of sakata - cart). When other naksatras are pierced or entered by graha, it is confirmed by seeing with instrument. Shapes of naksatras and planets moving north or south (beyond naksatra) can be seen in Brhatsamhita by Varahamihira. Thus positions of many stars have been told which are famous since ancient times. There are many other stars in unlimited number. Nothing has been told here about naksatras expect asvini etc. Notes (1) The graha move in ecliptic with little deviation according to their small sara. Many naksatras have large deviations, where the graha Conjunction with Stars 677 will never reach. Punarvasu is lying on ecliptic, hence it is crossed by all planets. This was the naksatra which determined start of solar year and malamasa in lunar year in vedic era. Hence its name is punarvasu, i.e. resettle or restart of year. 13 naksatras with less deviations can be crossed by planets. Sakata bheda - Rohini is in shape of cart i.e. sakata, hence its bheda is called sakata bheda. Its yogatara has 5° 32' south sara, but northern most star has 2° 35' south sara. According to siddhanta darpana it is 2° 20'. Moon has sara up to 5° hence it can easily cross rohini, but except budha and sukra, no other graha has parama sara of this value. Parama sara of sani is 2°29'39" hence sanis sakata bheda also appears impossible. But Varahamihira and Grahalaghava author have stated that sakata bheda by sani or mangala is very inauspicious. For siddhanta darpana value of sanis sara, its sakata bheda is just possible (at 2° 20' south sara). Mangala parama sara is only 1°51' according to siddhanta darpana and modern value but still less according to earlier texts. Vedha of rohini is possible only when south sara is assumed less, which is not given in the texts. According to star catelogues 3000 to 6000 stars only can be seen with naked eye. There are about 10 11 stars in our galaxy (of average size of sun) and there are about 10 9 ) galaxies in universe. Verses 88-92 : Milky way - A fine circular way of dense fine stars in seen in the sky. This is called chaya patha, vaisvanara 678 Siddhdnta Darpana patha or abhijit marga (akasa ganga also). It is proposed to describe it. This chaya patha is circular. It crosses ecliptic in sayana karka and sayana makara beginning. Again it extends 60° north from sayana mesa to 63° south from sayana tula. This crosses south part of punarvasu and goes southwards. Then it crosses mula and sravana naksatras and goes upto centre of abhijit and sravana. From there, it goes northwards. From beginning point of karka, it goes north in two branches. This can be easily shown by diagram on a sphere. But in sky, it is seen half only at a time, hence it is impossible to show it. We can easily see the stars (separately) of milky way with telescope. We can also see pusya naksatra, black spots on sun, water, mountains and trees on moon. Telescope can show phases of budha and sukra also like moon. Ring around sani and many new planets and satelites can be seen with it. Notes : (1) The galaxy is called akasa ganga, chayapatha, visnupada etc. However vaisvanara patha is name of ecliptic according to many. The akasa ganga, is the disc portion of galaxy which is dense area with more number of stars, hence it looks like a way. The main portion of the galaxy is a disc of about 30 kilo per sec width. It has two spiral arms and sun is located in inner arm as shown in figure 4(a) and 4(b). Sun is 10 Kpc away from centre i.e. about 2/3rd of the radius. 1 persec = 3.26 light years approximately, kilo = 1000. The galaxy rotates along the central plane of disc, which is almost parallel to orbits of solar system, central Conjunction with Stars 679 portion rotates with uniform velocity almost like a rigid body. Stars in vicinity of sun in the disc are rotating with speed of about 220-250 kms/sec around galactic centre. Mass of galaxy inside sun's orbit is 1.4 X 10 11 sun masses. Total energy of galaxy also is about 0.8 X 10 11 of sun. Mass of stars is 2X10 44 gram. \« 30 Kilopersec Figure 4a - Structure of galaxy Figure 4b - Spiral arms The points in figure 4a represent some of the globular clustures. The position of sun is marked with the sign. Regions are marked 1 to 5 - 1. The spherical subsystem, 2 - the disk, 3 -' the nucleus, 4 - the layer of gas dust clouds, 5 - the corona. Radius of corona is at least a dozen time the radius of galaxy. Figure 4(b) is disc of the galaxy. The nucleus is at centre C. Two spiral arms are spreading from it. Sun is in one of the arms. ' Spread of galaxy can be seen from figure 4c. C is centre and E is edge of disc. Sun is S. So SC = 10, CE = 15 kpc. ^.ESC=0 is spread of disc. Tan0 = 3/4 hence = 60° ap- proximately. Figure 4C - spread of galaxy disc 680 Siddhdnta Darpana Central portion and the disc are dense and obscured by gases. It can be observed only by radio telescopes. It is believed that nucleus of galaxy contains huge black hole. Spherical subsys- tem contains old stars and globular clustures. They rotate with about l/5th velocity of disc stars. Mass of corona is many times the mass of galaxy, but its density is much less and it does not emit any light. It is felt only by its gravitation. Centre of galaxy is in mula naksatra. its old name was mula barhani - i.e. the root from which cosmic egg has spread. Probably its position as galectic centre was known. Linga purana also states that brahma travelled for 30,000 years in cosmic siva linga - this is the distance in light years from centre to sun. ' . ■ (2) A note about magnitude of stars - The visual magnitude of the stars has been made in a logarithmic scale. Star of 1st magnitude is 100 times bright than 6th magnitude i.e. increase of 5 magnitude reduces the brightness by 1/100. Change in magnitude of 1 reduces the brightness by (100)' 1/5 = 1/2.5 approx. Brightness in this scale is Sun - 26.5 i.e. 6,31,000 times moon Moon - 12.0 Venus - 3.0 Sirius -1.4 Rohini + 1.0 Brahmahrdaya 0.1 Absolute magnitude is measured by emitting power compared to sun in similar logarithmic scale. Conjunction with Stars 681 Verses 93-94 : Prayer and conclusion May supreme lord Jagannatha destroy our forest of mishaps, who puts the golden ornaments to shame with his yellow dress, who is closely watching the creation and events in the cosmic egg, who is expert in dancing on hoods of Kaliya naga and who is radiant near tree of desires. Thus ends the twelfth chapter describing conjunction of graha and naksatra in siddhanta darpana written as text book and correction of calculation by Sri Candrasekhara born in famous royal family of Orissa. Chapter - 13 RISING SETTING OF PLANETS, STARS Graharksodayasta samaya varnana Verse 1 - Scope - Now I describe the rising and setting of planets and stars. In sphutadhikara, udaya and asta have been roughly described on the basis of difference of their kendramsa. Verses 2-3 : Types of rising and setting - Udaya and asta are of two types - Nitya (daily) and naimittika (occasional or seen). In first nitya type, due to rotation of pravaha. (daily rotation of earth), planets and stars rise daily in the east and set in the west. Hence, it is called nitya (regular) or pratyaha (daily). The planets rise when they are far from sun and are visible. They set when they become invisible due to closeness of sun. This is called naimittika (i.e. casual) udayasta. Verses 4-6 : Rising setting of planets (Surya siddhanta) - Mangala, guru and sani, set in west when their longitude (rasi etc.) is more than sun, when it is less then sun, they rise in the east. Vakri budha and sukra also set in west and rise in east, when their longitude is more than sun or less than it. Here more and less do not mean numerically bigger rasi. If the planet is ahead of sun in nearer portion of arc, then it is more in rasi and if behind, Rising Setting of Planets, Stars 683 it is west. For example, sun in mesa will be considered more than a planet in mina, because mesa is unmedately after mina. From mesa to mina direction, mina is greater in numbers but it comes at the end of circle. (Surya siddhanta) - When budha and sukra, moon are less than surya, they set in east. When they are more than surya, they rise in west. This is because budha and sukra are faster than sun. Notes : In general the rising and setting of planets etc is due to daily rotation of earth, due to which each star rises in east and sets in west. This is called daily rising and setting. Siddhanta darpana assumes that earth is fixed and the sky is rotated east to west by a wind pravaha, which is equivalent to daily rotation of earth. This chapters deals with the other type of rising and setting caused by brightness of sun. In western astronomy, it is called heliacal rising and setting (heliacal = caused by sun, Helios = sun in Greek). In this rising, when the planets are very close to sun and they rise around sunrise in east and set with sun, they cannot be seen due to closeness of sun. They are said asta (naimitika) or heliacally set. When they are slightly away from sun and are seen slightly before sun rise (in east or west) or after sunset, they are considered heliacally risen or naimittika udaya. First part of the discussion is about mangala, guru and sani which are slower than sun. When sun is behind them sun appears to be moving towards them. When they become very close, trlese planets become invisible. Before that closeness, they 684 Siddhanta Darpana are seen after sunset in west. After some days, they become invisible due to closeness of sun, hence they are said to set (heliacally) in west. After the time of closeness, sun goes ahead, then the planets are seen in east before sun rise. Hence the three planets are said to rise in east (heliacally). When vakri budha and sukra are ahead of sun, then they are seen in west after sunset and set there itself. After some days, they go to the other side of sun (less longitude, or west) and they are seen in east before sunrise. Hence vakri budha and sukra set in west and rise in east. When margi budha, sukra (and candra) are behind sun, they become nearer due to more speed and become invisible due to closeness. Then they are behind sun and are invisible in east before sun rise. Hence they heliacally set in east when they go ahead of sun, they are visible in west after sun set and are said to rise in west. Verse 7-11 : Drkkarma for rising and setting - For knowing the rising or setting time of a graha in west, on the approximate day of rising or setting, spasta surya and graha are found at sunset time. If the rising or setting is to be calculated in the east, then on apporoximate day they are calculated at sunrise time. (The ap- proximate time of rising or setting is known from rough kendrarhsa as explained in spasta-dhikara). After that drkkrama of both the planets is done. (Surya siddhanta quotation). Ayana drkkarma is done first, then aksa drkkarma is done. Method for aksa drkkrama - Sphuta sara of graha in kala is multiplied by palabha and divided Rising Setting of Planets, Stars 685 by 12. Result is multiplied by 1800 and divided by lagna asu of that time. Result will be in kala etc. This will added for south sara of graha and substracted for north sara when sun is in east horizon. When sun is setting in west, reverse order will be followed. (Surya siddhanta) Difference of aksa karma corrected graha and sun in asu is divided by 60 to find kalamsa. For rising and setting in west, we find the difference between (6 rasis added to graha) and the sun. By correcting graha with that, we again find kalamsa difference. Nati correction in sphuta sara of moon is done by the method explained in surya grahana (chapter 9). To see the setting of moon in east, it is added to accurate moon and substracted from it to see the rise in west. p N A Figure 1 - Aksakarma at Ksitija Notes - (1) Aksa drkkrama at Ksitija - In Fig.l NALS - Eastern horizon N = North point S = position of rising graha L = Udaya lagna 656 Siddhdnta Darpana K = Kadamba, P = pole B = position of rising planet S on kranti vrtta C = Planet S on kranti vrtta on dhruva prota circle. ABD = Diurnal circle of B CL = Aksa drkkarma of C. Here the planet with south sara, rises after its ecliptic position B has risen, or dhruvaprota position of C has gone further above. Diurnal circle of B cuts, dhruva prota of S on D. ADS is a spherical right angled triangle, because AD is parallel to equator, hence perpen- dicular to dhruvaprota line. Hence L DAS = lambamsa = 90°^>, where is aksamsa, /.DSA=0 Small triangle DSA can be considered a plane figure * Hence DA = Sin Z. DSA _ Sin <i> _ Pabbha DS ~ Sin L DAS " Sin (90° - <J>) 12 Now, approximately SD = SB and DA = CL CL _ DA palabha BS ~ DS = 12 ^ Sara x palabha or CL = ^ , as BS = Sara Since in south sara the planet is above horizon at sun rise time, its aksa correction is added to the planet. (2) Kalamsa is the time before sunrise when a planet rises. It is equal to 1 asu for 1 kala difference on equator. The difference between sun Rising Setting of Planets, Stars 687 and planet corrected for aksa karma wil be rising time difference in asu = Kalantara in kala , , . . . , Kalantara in kala Hence, kalantara in amsa = — Rising time diff in asu = '~ 60 (3) Nati samskara is needed only for moon as it is very little for other planets. Verse 12 - When the larhbana corrected moon is at 11° kalamsa from sun, then it is seen on horizon. Wh?h its distance is less than 11 kalamsa it cannot be seen. Verses 13-16 - Rising of stars Sara of naksatra are 'bigger. Hence sphuta kranti is found from their sara. For this sphuta kranti, carajya and day length in asu is found. That will give daily rising or setting time and lagna. At the time of rising (or setting), we get the difference of lagna and sun. The rising time for that difference in asu divided by 60 will give kalamsa. The kalamsa at the time of rise in east or setting in west is only dependent on sun motion, because stars don't move. Hence, they rise or set at distance of kalamsa from sun in east or west like farther planets mangala etc. Before setting in west the stars rise in east, due to daily motion. It is not connected to distance from sun. Notes : The method explained earlier for grahas was approximate for small sara. But naksatras have bigger sara and accurate method as 688 Siddhanta Darpana explained in chapter 11 - for conjunction of planets is to be used. The rising time difference is found by sphuta kranti of star and sun i.e. it will be difference in their carajya only. Since it is at times of sun set or sun rise, proportional difference for natamsa of sun need not be made. _. , . Carjya difference in asu Thus, kalamsa = — Verses 17-24 - Kalamsa of stars Kalamsa of naksatras in degrees depends on their birhba diameters in vikala. The observed values of kalamsa for successive rise in birhba vikala is given below - Birhba Vikala Kalamsa 1 24 1/15 23 1/30 22 1/45 21 2 20 2/30 19 3 18 4 17 5 17 6 16 7 16 8 15 9 15 ,10 15 11, 12, 13 14 15, 16 13 17, 18, 19 12 20, 21, 22 11 Rising Setting of Planets, Stars 689 23 to 28 10 29 to 40 9 Notes : Bimba value is not the diameter of stars because it is so small, it cannot be measured even with telescope. It is only a measure of brightness estimated empirically. Actually the visibility distance (kalamsa) from sun is one of the measures of brightness - expressed as bimba diameter. Verses 25 - Kalamsa of tara grahas Kalamsa of taragrahas are Sukra 9, vrhaspati - 11, budha 13, sani 15, mangala 17. For sukra and budha, the above values are average. Their kalamsa at cakra or cakrardha (0° from sun - farther side is cakra, 180° from sun i.e. near side is cakrardha) are Cakra Cakrardha Budha 14 12 Sukra 10 8 Note : At the end of cakra, on farther side of sun, the planets are farther hence light intensity is smaller. Hence, they become invisible at greater distance. Then they rise in east and set in west. At cakrardha, budha and sukra are between earth and sun and vakri, then they set in east and rise in west. Verse 27-29 : Rules for heliacal rising When difference between sun and the planet or star is more than the kalamsa, it will not be visible (due to light of sun). 590 Siddhanta Darpana Difference of graha or naksatra with sun being less than its kalamsa in west, means it has already set. If difference in more than kalamsa, then it will set in near future. If in east direction, rising will be in reverse order. If difference (kalantara) is more than kalamsa, then planet has already risen, if less than kalamsa, it is yet to rise. Notes - (1) Condition of rising are (1) Planet should be above horizon. (2) It should be night time for its being visible. Even in night, slightly before sunrise or after sunset, it becomes invisible due to twilight. In the limiting case of rising in east, its difference from sun should be more than kalamsa. Bigger or brighter planet will be visible at lesser distance from sun. (2) Setting in west - slow planets mangala, guru or sani or stars are overtaken by faster sun. In west, they rise after sunset at the minimum distance of kalamsa, when it has become sufficiently dark. It the limiting case, they are east from sun at kalamsa distance, when distance is more, it will be reduced in future, when the planets or star will set. Same happens with vakrl budha or sukra. For rising at sunrise time, they should rise before sunrise, i.e. west from sun at kalamsa. Distance of sun increases due to its faster speed in east direction. Hence if it is more than kalamsa, it was equal to kalamsa earlier, when planet or star has risen. Vakrl budha and sukra also are separated further as they are in west and moving further west. Rising Setting of Planets, Stars 691 Verses 30-33 - Day of rising or setting of planets - Graha and sun are added with ayanamsa. For setting time, six rasi is added to both. For graha and sayana sun at rising time (6 rasi added to each for setting time) rising time of their rasis are multiplied by daily speed and divided by (1800). Result will be kala gati at the time of rising or setting. We calculate the difference between rising times of the planet and sun at sunset or sun rise times before aksakarma. This is ista kalamsa from * * which kalamsa of rising is substracted. This kalantara is divided by difference of kalagati of sun and forward moving graha. If graha is retrograde kalantara is divided by sum of kalagatis. Result will be past days or coming days of rising or setting, as per rules explained earlier. Notes : (1) Since inclination of planetary orbits with ecliptic is very small, it can be assumed to move on ecliptic. 1800 kala on ecliptic = rising time of that rasi on equator is asu ~_, ^ . ^ , gati kala x rising time or Kala gati in 1 day = - - - (1) 6 J 1800 v } (Kala gati in kala.) (2) Kalamsa antara or kalantara = Kalamsa of graha on ista day - parama jfcalamsa of graha i Past or remaining days %■ Kalantara = — (2) v Sun kala gati ± graha kalagati v ' ,, Here + sign is for vakri graha and - ve sign *s for forward graha. (3) 692 Siddhanta Darpana Verses 34-37 - Aksa drkkarma for stars. Difference of ista kalamsa between naksatra and sphuta ravi and the parama kalamsa is divided by kala gati of sun at the place of naksatra dhruva. This will give past time or remaining time for udaya kala (if sphuta sun at rising time is taken) or asta kala. Half day of naksatra is calculated for sphuta kranti and asphuta kranti. Their difference in asu is multiplied by 1800 and divided by rising time of the rasi for rising in east (or rising time of 6 rasi + naksatra for setting in west).' Result will be aksa drkkrama in kala. When naksatra has north sara, aksa kala will be added to naksatra in west and substracted from naksatra in east. For south sara of naksatra, aksa kala is added to naksatra in east and substracted for naksatra in west. Result is drkkarma corrected dhruva and ksetrarhsa is found from that. Notes : (1) Naksatras have no proper motion, hence their rising time is calculated only from sun's motions. Here, in place of sun gati ± naksatra gati = sun gati - = sun gati only. Similarly, kranti of naksatra is fixed, hence ayana drkkarma is not done, only aksa karma is done. (2) Udaya lagna or udaya vilagna of a star is that point of the ecliptic which rises in the eastern horizon simultaneously with the star and the asta vilgana or asta lagna of a star is the point of ecliptic which rises on the east horizon when the star sets on western horizon. Rising Setting of Planets, Stars 693 z :.jc Figure 2 - Aksa drkkarma of a naksatra Figure 2 is celestial sphere for a place of latitude O., SEN is horizon, S, E, N being south, east and north cardinal points. Z is zenith. X is position of star when it rises on the horizon (eastern). TEB is the equator and P its north pole. TLA is the ecliptic and L is the point of the .ecliptic which rises with star X i.e. star's udaya lagna. The point T where the ecliptic intersects the equator is the first point of Aries (sayana mesa). PXAB is the hour circle (dhruvaprota circle) of star X and A the point where it intersects the ecliptic. U is the point where diurnal circle of A intersects the horizon. Now arc EB is the ascensional difference (carajya) due to true declination (arc XB) i.e. spasta kranti of star. Arc EM is carjya due to madhya Ikranti (of the star's position on ecliptic) i.e. arc AB of the star. The difference of these carajya is arc MB in asu. In asu of arc MB, portion CA of ecliptic nses. CA has been approximately considered equal to LA, aksa drkkrama of star. 694 Siddhanta Darpana Thus aksa drkkrama = Carajya for spasta kranti - carajya for madhya kranti. Since dinardha = 15 ghatl + carajya difference of dinardha = diff. of carajya - (1) = aksa drkkarma Longitude of the star's udaya lagna L i.e. arc TL = arc TA - arc LA = arc TA - arc CA approx = Polar longitude - aksa drkkarma. (2) This explains when the star is north of the ecliptic, then aksa drkkarma is substracted from star to find udaya lagna in east. For asta lagna, it will be added to polar longitude of star (dhumvamsa) added to six rasis. (3) Rules for rising and setting can be stated as star rises heliacally when sun's longitude = udaya lagna of star + kalamsa. It sets heliacally when sun's longitude = astalagna - kalamsa - 6 signs. Star is invisible if, Sun's longitude - udaya lagna < kalamsa or, asta lagna - sun's longitude < kalamsa + 6 rasis This can be stated in terms of udayarka and astarka. Udayarka (or udaya surya) is position of sun when a star rises heliacally. Astarka is position of sun when a star sets heliacally. Calculation of udayarka — Star's udaya lagna is taken as sun's longitude and it is assumed that Rising Setting of Planets, Stars 695 time elapsed since sunrise is equal to star's kalamsa ghatis. Lagna at that time is itself udayarka. Calculation of astarka - Star's asta lagna is taken as sun's longitude and kalamsa ghatis of time before sunrise, we calculate the lagna. By adding 6 rasis to this lagna we get astarka. Theorem (1) - If astarka > udayarka, star will never set. When sun = udayarka, the star rises heliacally. Thereafter, as the sun moves, distance of sun from udayalagna increases and star remains visible. Since astarka > udayarka, the same happens when sun = astarka. The star therefore, does not set when sun = astarka. Thus setting is impossible in this case. This happens, when star has sufficiently big north latitude (for places of north latitude), such that star's aksa drkkarma > star's kalamsa (on ecliptic). For, in the case. Udayarka = star's polar longitude - aksa drkkarma + kalamsa < Star's polar longitude and, astarka = star's polar longitude + aksa drkkarma - kalamsa > star's polar longitude So that, Astarka > star's polar longitude > udayarka Theorem (2) - If, for a star, astarka < udayarka, then the star will rise and also set. The star will remain set when, astarka < sun < udayarka and will remain visible when sun < astarka but > udayarka. 696 Siddhdnta Darpana Proof - When sun = astarka, the star sets heliacally. As the sun's longitude increases, the distance between asta lagna and sun diminishes and star remains heliacally set. This happens until sun = udayarka, when the star rises helically. Thus sun remains set until, astarka < sun < udayarka. When sun goes beyond this limit, it is helically visible. This case happens when the star's latitude is north and its aksa drkkarma < kalamsa of star. For, udayarka = polar longitude of star - aksa drkkarma + kalamsa on ecliptic > star's polar longitude and, Astarka = Star's polar longitude + aksa drkkarma - kalamsa < star's polar longitude so that, astarka < star's polar longitude < udayarka This also happens, when star's latitude is south. For, udayarka = star's polar longitude + aksa drkkarma + kalamsa * > star's polar longitude Astarka = star's polar longitude - aksa drkkarma - kalamsa < star's polar longitude so that, Astarka < star's polar longitude < udayarka. Rule for set period : A star remains heliacally set until astarka < sun < udayarka Rising Setting of Planets, Stars 697 Between this period we take sun's speed as the speed at position of star's polar longitude which is in between these two values, hence can be taken as average speed. Hence this period for setting Udayarka - Astarka Average speed of sun Verses 38 - 40 : Kalamsa of the yogatara of a naksatra is expressed in kala, multiplied by 1800 and divided by rising time of its sayana rasi (for rising) and by rising time of (sayana rasi + 6 rasis) for setting. Result will be ksetramsa in kranti vrtta. This will be added to drkkarma dhruva of naksatra for rising or substracted for setting. This will be udaya or asta dhruva of yogatara. When sun's dhruva (polar longitude) is equal to udaya or asta dhruva of the yogatara, it will helically rise or set. Notes : It has been explained is previous note. Udaya dhruva is udayarka, asta dhruva is astarka. Verses 41-44 : Extreme cases - Manj naksatras in north rise again in east before they set in west. Hence their setting is not necessary. Their setting has been discussed only to know the setting time in west. Udaya and asta of many naksatras like Kratu should be calculated. Agastya and yama are in far south, hence they remain set for long. Day length of any graha or star can be known from its carajya calculated from kranti (and local aksarhsa). Hence, their daily rising and setting times can be known. Still, detailed methods will be explained here for their times of udaya and asta. The discussion so far has been done according to the views of earlier astronomers. Now I describe more accurate methods thought by me. 698 Siddhdnta Darpana Notes (1) Permanent rising and setling of stars has been explained earlier. If kranti of the star is more than colatitude of the place, the star will never set for north kranti. For south kranti, greater than colatitude of the place, it will never rise. Equivalent condition is that, astarka > udayarka; i.e. star will rise again before it sets in west, explained in theorem (1) of previous note (3) after verse 37. (2) Rising of agastya (canopus) has been discussed extensively. According to Aryabhata I, Varahamihira and Sumati, agastya rises heliacally when sun's longitude = 120° + <J> and sets heliacally when sun's longitude = 180° -(120° + O ) = 60° -O where O is the latitude of the place. According to Vatesvara; agastya rises heliacally when sun's longitude = 98° + 42 P/5 degrees and sets heliacally when it is 76° - 42P/5 degrees. where P is the equinoctical mid day shadow in angulas. Manjula gives the formulas as 97° + 8P and 77-8P. Bhaskara II and Ganesa daivajna give 98°+8P and 78° - 8P The above rules have been derived by substitution from the following formula Rising Setting of Planets, Stars 699 Udayarka = star's polar longitude + aksa drkkarma + kalamsa. Astarka = star's polar longitude - aksa drkkarma - kalamsa. * Verses 45-50 - Sphuta kalamsa - Planets and stars rise on horizon, when sun is still below horizon. Even in such situation they are invisible because sun's light reaches on horizon (twilight) due to reflection from atmosphere. Natarhsa of sun from drk - mandala (when it is start of twilight) is multiplied by trijya and divided by lambajya. Result is again multiplied by trijya and divided by dyujya. Result will be sphuta kalamsa of stars from sun. This means that, there is big difference between drk-mandala arhsa and kalamsa. On equator also, it is equal to difference between dyujya and trijya. At other places it is still more. For example at a place of 66° north aksamsa, in mesa month (when sun is in mesa rasi) guru and sukra in mlna rasi rise alongwith sun. Both these planets are seen only when away from sun. Hence, it is not necessary to calculate these kalamsa difference in rising times. From the kalamsa written for these planets, ksetramsa is more, though kalamsa is below the visibility limit. Hence, they are seen. Notes : Due to reflection from atmosphere, twilight starts when sun is still 18° below horizon. In India, it is assumed 15° below horizon, as it is north of equator. This is called usa in morning 700 Siddhanta Darpana and sandhya is evening. Sandhya is used for both twilight periods. Then sun rises when it is still about 35' below horizon due to refraction of rays in atmosphere. Hence twilight period extends from 18° below horizon to 35' below horizon position of sun. Thus the natamsa of sun below horizon (18°) or natamsa of 108° from meridian is the time when sun light starts. Thus, it is increase in carajya which is equal to increase in half day length. Like carajya, the natamsa jya is divided by cos <p = lamhajya/trijya to find rising difference on diurnal circle. It is divided by cos d = R cos d /R = dyujya/trijya to get the degrees on equator whose kala is equal to asu time. Hence the formula. Here, dyujya on equator means kotijya of natamsa, instead of kotijya of kranti. For 66° north aksamsa, the difference is Sin 18° /cos 66° = Sin 30° approx. Hence guru and sukra rise with 30° or 1 rasi difference. Verses 50-58 - Sphuta dhruva of udayasta of graha - From the udaya and asta kendramsa, we find the udaya and asta kala of planets. For that time, mandaphala of sun is calculated. This mandaphala is substracted from fourth sighra kendra of guru, mahgala, sani at the time of udaya or asta, or added to it in same manner, it is substracted or added to sun. In budha or sukra, this correction will be in reverse order. If graha is less then the true sayana sun at that time, half the kalamsa of graha is substracted from sun. If graha is more than sayana sun, then half kalamsa is added to sayana sun. If sun is in I Rising Setting of Planets, Stars 701 east, that will the lagna at that time. When sun is in west, 6 rasi is added to sun ± half kalamsa. That will be the lagna for setting time. 3 rasi is substracted from this lagna. Kranti of that point of ecliptic (tribhona lagna) is calcu- lated. By adding or substracting aksamsa to kranti, natamsa and unnatamsa is found (for tribhona lagna). The natamsa of sukra, guru, budha, sani are divided by 4,5,6,7,8 and result is added to unnatamsa. Jya of the resulting angle is found. Kalamsa of the planets for udaya or asta is multiplied by trijya and divided by jya of the corrected unnatamsa. Result is degrees etc. will be kalamsa of graha in ecliptic. This substracted from sun will be dhruva for asta or udaya. Half of this asta or udaya dhruva, is added or substracted from sayana sun as before - That will give corrected udaya lagna or asta lagna. Notes : No logic has been given for such a long and arbitrary process. Probable justification is given below - (1) Kalamsa difference from sun is measure of decrease in intensity of sun light. Since it decreases according to inverse square of distance, kalamsa proportinate to birhba diameter (measure of intensity) is reduced by half. (2) Mandaphala substracted from sighra kendra, is distance of planet from sun, on which tile brightness of graha depends. (3) Natamsa of tribhona lagna is proportional to inclination of diurnal circle with vertical. The kalamsa will increase in proportion to this obliquity. 702 Siddhanta Darpana It is divided by half the values of kalamsa of graha. For bright planet, kalamsa is less, fraction of natamsa is more, then corrected unnatamsa and its jya will be more, hence sphuta kalamsa will be less as it is divided by jya. Still the method appears to be based on trial and error and probably gave better results. Verses 59-68 : Ksetramsa of planets for mid Orissa Ksetramsa of planets is being given for mid Orissa according to rasi of sayana sun. Planet Rasi of sayana sun ksetramsa Sukra 1, 12 2, 11 3, 10 4, 9 5,8 6-7 10/26 9/51 9/40 9/20 9/0 9/1 When sun is in west, 6 rasi is deducted from it and then ksetramsa is found. Then the degrees of sayana sun are multiplied by difference of dhruvamsa and added to ksetramsa if increasing, or substracted if decreasing. Guru Budha 1, 12 13727' 2-11 1372' 3, 10 12°11' 4, 9 11/28 5, 9 1175 6, 7 11/0 1, 12 16/11 2, 11 15/39 3, 10 14/32 4, 9 13/36 Rising Setting of Planets, Stars 703 Sard Mangala 5, 8 6, 7 1, 12 2, 11 3, 10 4, 9 5, 8 6, 7 1, 12 2, 11 3, 10 4, 9 5, 8 6, 7 13/7 13/0 and 13/1 18/56 18/26 16/53 15/44 15/8 15/0, 15/1 21/42 20/53 19/14 17/52 17/10 17/0, 17/1 In fifth rasi, astodaya dhruva will be equal to madhyama ksetramsa. Difference of 6th and 7th rasi has been written as 1' kala. Verse 69 : For other places also, from unnatajya, ksetramsa between sun and the planet for udaya or asta can be found out. Verses 70-76 - Sphuta udayasta time. When difference of sun and graha is equal to dhruvamsa, then that will be the sphuta time. It will be made more correct by successive approxima- tion. From sara of graha or naksatra, both ayana and aksa drkkrama correction are done. Both; corrections are added together or difference is taken according to sign. Resultant correction (positive or negative) in vikala is divided by difference of sun gati and graha gati in kala. When budha or sukra are vakri, it will be divided by sum of gati kala. Siddhanta Darpana Result in danda etc. is added to rising and setting time, if rising is in east and setting in west - and when drkkarma was positive. If rising is in west and setting in east, then it is substracted from rising or setting times. For negative drkkarma, reverse in done. The correction time is substracted from rising time in east or setting time in west. It is added to rising time in west or setting time in east. Then we get more correct time for udaya or asta. This correction is due to motion of planet at the cakra or cakradha. When sphuta sun and sphuta graha are in same rasi then it is cakra time for all tara graha, for budha and sukra it can be cakrardha also. Note - Ayana and aksa drkkarma have already been explained in conjunction of planets. Positive ' drkkarma means rising of planet is later and setting time is earlier, i.e. difference with sun i» reduced, (rising in east and setting in west). Then their difference will again increase to dhruvamsa distance after sometime depending on relative speed. Verses 77-82 - Astodaya time without drkkarma - For Orissa, udayasta degrees for eacn pianet has been stated according to sayana sun in each rasi. At the time of cakra, the degrees ot udayasta are divided by difference of speeds ; ot sun and graha. For cakrardha of budha and _sukra, division is by sum of gatis. Result will ^e time ui days etc. For that period after cakra or cakrardha graha will be set, then it will rise. It will be set for that period before cakra/cakrardha also. u &■■■ Rising Setting of Planets, Stars 705 Sayana sun is calculated for the time of udaya or asta found approximately. Again, we calculate the difference of sun-graha distance and ksetramsa. This is divided by gati antara or sum and udayasta times are corrected. For this corrected udayasta kala, we take the average position of graha and sun. Speed of sun and graha for that position is the sphuta gati for both for purpose of udaya or asta times. At any time we calculate the difference of sayana and dhruvamsa corrected sun and the sphuta graha. That is converted to kala and divided by difference or sum of speeds of sun and planet. That will give days since udaya or asta or remaining days according to rules explained earlier. Notes : (1) At cakra and cakrardha planet has same position from earth as sun. Then they will be set due to closeness of sun. Assuming the speed at end of cakra/cakrardha to be average speed up to period of udaya, we calculate the time when the planets will be separated at distance of ksetramsa, %hen they will rise heliacalry. By successive approximation by speeds at approximate time of udayasta, we calculate more accurate time. (2) The days since udayasta or remaining days are calculated by calculating as to when sayana Sun - sayana graha = ksetramsa. (3) 1/2 (sun + graha) at udaya or asta time is the mid position of sun and graha. With sufficient accuracy, speeds of sun and graha at that position can be considered sphuta. 706 Siddhanta Darpana Verses 82-84 : Conclusion - Other astronomers have stated the udayasta degrees of graha 1° less than values stated here. According to them the degrees are - guru 10, budha 12, sani 14, mangala 16, vakrl sukra 7, margl sukra 9. This is not the real setting or loss of a planet. In course of rotation of earth and their own motion, they keep coming east, west, up or down. As eyes are dazed due to brightness of sun, tara graha become lightless like light flies and become in- visible. Due to bigger angular diameters, candra, guru and sukra are seen in day light also. At equator 22° 30' kalamsa before rise or after setting of sun, its light starts reaching horizon. At other places this kalamsa is multiplied by lambajya of the place and divided by trijya. Light of sun will go upto that distance, (kalamsa). At equator, light of moon is visible 8 kalamsa after, setting or before rise. Light of sukra is visible 1 kalamsa before rise or after setting. Kalamsa at other places is obtained by multiplying it with trijya and dividng with lambamsa jya. Notes : It ha,s already been explained, how planets set hetically. Kalamsa of sun here has been taken as 22° 30' at equator against 18° taken in modern astronomy. Kalamsa of moon and sukra is not calculated, as it is ineffective compared to light of other stars. At other places, kalamsa depends both on kranti and aksamsa as calculated earlier. Roughly we can assume that, sun rays will reach same Rising Setting of Planets, Stars 707 manamsa at other places also which is equal to Icalamsa X trijya / lambajya Verses 85-86 - Prayer and end I pray thousand times to revered lord Jagan- natha, whose brightness is like jewel of Indra (Indra nila mani is blue), whose lotus feet are worshipped by Vasuki, Ganesha, Siva, moon and sun. Thus ends the thirteenth chapter discribing rising and setting in Siddhanta Darpana written for correct calculation and a text book by Sri Candrasekhara of a famous royal family of Orissa. Chapter - 14 LUNAR HORNS Candra Smgonnati Varnana (Elevation of lunar homs) Verse 1 - Scope - For knowledge of persons of sharp intellect, it is proposed to describe accurate daily rising and setting of moon, elevation of lunar horns (candra srnga) and diagrams (parilekha). Verses 2-11 : Time after sun set when moon sets. Rising time and setting time of moon are Calculated roughly according to method described earlier. At the time of sun set, accurate moon and sun are made sayana (ayanarhsa is added). Then drkkarma sanskara is done. 6 raSi is added to sayana and drkkarma corrected sun and moon. By difference of their lagna (rising times of rasis between them), kalamsa is found out. Among sun and moon, bhogya (remaining) asu of lesser rasi, bhukta (lapsed) asu of bigger rasi and rising time, Of other rasis in between in asu - all are added and divided by 60. Result wiU be sphuta kalamsa difference between sun and moon. Kalamsa difference divided by 6 gives the result in ghati etc. This is multiplied by gati ot sun and moon and divided by 60. Result in kala lunar Horns 709 etc is added to sun and moon. Again, we find the difference of their rising times. After repeated procedures, rising time difference between sun and moon will become steady or fixed. Here, ayana and aksa drkkarma of moon is to be done every time, otherwise sara will be different due to change in distance between moon and its path. When the rising time difference between moon and sun becomes constant thus, 3 rasis are substracted from moon. For this vitribha lagna, nata and sara are found, and its ayana and aksa drkkarma are done. To make it more accurate, the drkkarma correction is made to moon and sun with 6 rasis added to them. The rising time difference in asu is found. Sphuta gati of moon is divided by 14 and multiplied by trijya and divided by lambajya to get the lambana asu of moon. On substracting this from the rising time difference, we get correct difference in asu. This period after sun set, moon will set. While finding instantaneous sun and moon, asu should fee considered savana (21,659 part of savana day) and while finding moon at any time it will be staksatra (i.e. 21600 part of a naksatra day). At sun set time, asu will be candra savana (i.e.. 22, 390 Notes : (1) Second half of fifth and 1st half ff sixth verse are quoted from siirya siddhanta, Which was considered by many to be an interpola- tion. However, here, they are further specified by the values of asu to be taken in these 710 Siddhdnta Darpana calculations. This is the method for calculating setting of moon in sukla paksa (bright half). Though this is not specified anywhere, but next verse tells about procedure for krsna paksa. It has been clearly specified in surya siddhanta. (2) Rough method for rising and setting time of moon - This has been stated in chapter 8 - candragrahana verses 60-65. That is for purnima and can be used for 8th of sukla paksa to 7th of krsna paksa. This ignores sara and doesn't do drkkarma sanskara. Rising time - At sun set time sayana sun and moon are calculated. Rising time of moon after sunrise = Rising time of remaining part of rasis of sayana sun + for lapsed part of sayana moon + for rasis in between sun and moon + 56 asu as lambana correction for moon = A asu = 360 ghatl Setting time of moon : Moon set time after sunset = rising time of remaining rasis (of sun at sun rise time + 6) + for lapsed part of rasi of (sayana moon at sun rise + 6 rasi) - 56 asu = A asu =^ghati. Sara correction - Sara kala X palabha / 12 is added to rising time if sara is south and substracted if sara is north. Reverse correction is done for moon set time. 711 f'-l', *'■■-' t M-, ' f I' ' lunar Horns Around purnima, moon rise is around the time of sunset, hence the position of sun and moon at sunset time are taken for better approximation. Rising time of moon - rising time of sun in east = rising periods of ecliptic between rasi of sayana sun to sayana moon. Due to larhbana, moon will appear lower when seen at horizon and rise 56 asus later or set 56 asus earlier, the time needed by moon to cover earth's radius in its orbit. When moon is setting (moon + 6 rasi) is rising in east. Similarly at sunset time sun + 6 rasi is rising. Hence moon set - sun set. = rise of (moon + 6 rasi) - rise of (sun + 6 rasi) = rising time between (moon + 6 rasi) to (sun + 6 rasi) For equator, rising time of a rasi and 6 rasi away from it is same. If we use the time of setting of rasis instead of rising, addition of 6 rasis is not needed. Sign Time of setting in asus 'at the equator at the local place 1. Mesa 1675 2. Vrsa 1796 3. Mithuna 1929 4. Karika 1929 5. Simha • 1796 6.Kanya 1675 Here a, b, c are the rising time differences for mesa, vrsa and mithuna. 1675 +. a 1796 + b 1929 + c 1929 - c 1796 - b 1675 - a Sign 12 Mlna 11 Kumbha 10 Makara 9 Dhanu 8 Vrscika 7 Tula 712 Siddhanta Darpana Due to north sara, effective kranti is increased, hence carajya will increase. Component of sara parallel to kranti i.e. perp to equator is s cos e where e is inclination of moon's orbit with equator. Hence, corresponding carajya increase is s x palabha s tan <I> = — — 12 This is substracted from rising time as day length increases due to increase in kranti. (3) Successive approximation and drkkarma - Due to aksa and ayana drkkarma, difference between sun and moon is corrected as visible from the place. Difference in moon set - sun set = . rising time diff (sayana moon + 6 rasi) (sayana sun + 6 rasi) as the rasi at 6 rasi difference is rising when sun or moon are setting. That will be sphuta time difference in asu. Asu Kala — — = —rzr = degree 60 60 6 360° = 60 danda (naksatra time) Hence 1 danda time = 6° Kalamsa Speeds of ravi and sun are calculated for savana dina, hence savana asu is to be used (1 day = 21659 asu). For calculation at sunset time, we take candra savana dina because moon set to moon set time is equal to candra savana dina. From speed of sun and moon at the asta time of moon, further corrections are done. &' |j<nar Horns 71 -3 (4) Lambana correction at setting time is sphuta gati of moon divided by 14. This is lambana amsa at local aksamsa. To convert it into kala at equator or asu, it is divided by cos <p i.e. R cos<£ / R or multiplied by trijya and divided by lambajya = R cos <p. This is substracted from setting time. Verses 12-13 '- In krsna paksa, sun at sunset time is calculated, 6 rasi is added to sayana sun. Difference in rising times between (sayana sun + 6 rasi) and sayana moon at that time is the time after sun set when moon wil rise. Here also drkkarma is to be done for sun and moon both at sun set time. At rising time, lambana asu is added. After repeated calculations with sun and moon positions at moon rise time we get steady value of diff ernce in sun set and moon rise in east. Notes : In krsna paksa, difference between sun and moon is more than 6 rasi. Hence at sun set time, moon is below east horizon, Hence, we calculate the difference between east horizon ecliptic point (i.e. sun + 6 rasi) and moon. However, while the position of moon at sunset time comes on ecliptic, moon goes further east due to its motion, hence real rising will be later. This difference is corrected by successive approximation. Verses 14-18 : Position of moon at desired time Now, method is described for calculating J position of moon at sun set time or any other time as observed from earth's surface. By method explained in chapter 6, sphuta candra is found at desired time. Its position east or west half of sky is found (from lagna etc.) 714 Siddhanta Darpana Lagna for desired time, vitribha lagna and vitribha sariku is calculated. Drgjiya for moon in east or west half of sky is multiplied by drggati (vitribha sanku) and divided by trijya. Result is multiplied by first sphuta gati of moon and divided by 14 X trijya 3438 (=' 48132). Result in kala etc will be added or substracted to moon, if moon is east or west from vitribha lagna. Then we get lambana corrected moon at desired time. After that, we find nati of moon again. Nati and sara are added or difference is taken according to same or different directions to get sphuta sara. From that, we do ay ana and aksa drkkarma. By making drkkarma correction, we get the samaprota vrtta moon as seen from earth surface. Notes : Methods of lambana and sphuta candra have already been explained in chapter 9 on solar eclipse. Verses 19-27 : Elevation of lunar horns There are two types of elevation of lunar horns (srnga). Generally horn means, pointed ends of the bright portion of the disc. But some authorities consider elevation of horns of black portion also. This horn is not seen but it can be known from calculations. * From one rise of moon to its next rising time is called savana day of moon. From sphuta kranti of moon, its nata kala is found by method explained earlier. In first half of sukla paksa (1st day to 8th) and second half of krsna paksa (9th day to 14th), elevation of bright horns is found. For other days i.e. 2nd half of sukla and 1st half of krsna paksa, Lunar Horns 715 elevation of dark horns is calculated. Out of bright or dark parts, whatever is less than half, its elevation is calculated. Moon in its vimandala (inclined orbit) is lighted by rays from sun in apavrtta (kranti vrtta or ecliptic) and is seen in many shapes. Even when both the horns of moon are equidistant from sun, they appear small or big and inclined. At a place where midday sun at the end of uttara-ayana (i.e in sayana mithuna) is above head (i.e. aksamsa of the place is equal to maximum kranti - karka rekha place), the sun at beginning of sayana mesa will be in sama mandala at sunset time (i.e. in east west circle). At that place moon with zero sara will move exactly in east direction. This shows that according to position of kranti vrtta, position, speed and horns of moon are decided. That also changes due to change of sara. As in candragrahana, in finding elevation of horns also, ayana and aksa valana are calculated. Verses 28-29 - Sara valana We consider the right angled triangle whose sides are (1) Perpendicular side is the jya of difference between moon and sun. (2) Base is the bhuja of sarajya (or sara with direction) (3) Square root of sum of these squares is karna - i.e. linear distance between sun and moon. 726 Siddhanta Darpana Sara is multiplied by trijya and divided by karna. Arc of the result in kala is divided by 60 to know the valanamsa of sara (i.e. angular deflection). Notes : Here we form the right angled triangle for deflection from ecliptic only. For sun it is zero. In surya siddhanta, it is calculated for defleciton from equator. For that we take the difference of kranti of sun and moon = p Then, base p x chaya karna of moon ± 12 aksajya lambajya Perpendicular = Sanku of moon i.e. kotijya of natamsa Then, Karna =Vbase 2 + perp 2 Here karna has been termed as madhyahna candra prabha karna i.e. straight distance (like a light ray) of mid day moon. This has been confused with mid day of sun. Ranganatha in his gudhartha prakasika tika on Surya siddhanta, interprated it as mid point of civil day between sun rise to next sunrise i.e. sunset time. Accordingly, he derived the formula. This was followed by Burgess who wrote the commentary in 1860 at Chicago U.S.A. after he got Ranganatha Tika in Maharastra in 1835. Svami Vijnanananda followed it in his Bangala commentary in 1909 and Sri Mahavira Pd Shrivas- tava in his Vijnana Bhasya in 1940. When moon is at meridian or its midday, sun is at horizon or above it i.e. within ± 90° of moon, as the bright portion is less than half for a horn Lunar Horns 717 to follow. Hence it will be almost correct for other positions also. Proof : SZ is a quarter of the yamyottara vrtta. Let Figure 1 C be its centre and Z zenith. Let EC be nadi mandala. ZC is produced to D, so that CD represents a sariku of 12 angulas. When sun is at E, DF is equinoctical shadow of CD or palabha. When sun is at A, DG called bhuja is shadow and GF agra. When sun is at B, DK called bhuja is shadow and FK agra. Thus Bhuja = Palabha ± Agra; or ± Agra, when sun is on horizon. Similarly bhuja of moon = Palabha ± moon's agra in the sphere whose radius is candra chaya karna i.e. hypotenus of right angled triangle whose one side is sanku of 12 angula and other is shadow caused by moon. Thus in this sphere, sun's agra Sun agra X candra chaya karna Moon's agra = Candra agra x Candra chaya karna Tnjya 718 Siddhanta Darpana tw aOT3 Kranti jya x Trijya But agra = . _ Lambajya Hence difference between sun's and moon's bhuja = Palabha ± (Candra kranti jya ± Sun kranti jya) Candra chaya karna Lambajya Palabha 12 , , . _ = ~ : — rr , hence we get the aksajya Lambajya ° formula for base. From this base and sanku of moon's height, we get the karna which is direction from sun to moon in meridian circle, i.e. projection of sun moon line in this circle. (2) Due to the confusion about this interpreta- tion and apporximate formula, siddhanta darpana has given more direct and accurate formula which can be used for any position of sun and moon. Figure 2 • It is known that sun is always on ecliptic, but position of moon at M on ecliptic is perpendicular foot on ecliptic. Thus MM' is perpendicular on plane of ecliptic, i.e. on line SM' of this plane also. Arc SM' is difference in moon and sun on ecliptic i.e. their rasi difference. Line SM' is the jya of that difference. MM' is sara of moon i.e. sara jya (arc MM' is the sara). Hence SM = ^$M' 2 + MM' 2 lunar Horns 719 gives prabha karna of moon at any time. (3) When we know the karna, at this distance sara gara will make an angle = radian sara x trijya kala . = - in kala karna ■ We are following the scale of 1 angula = 1' on khagola circle in diagram. Hence, kala is converted to degree or angula by dividing it with 60. Verses 30-43 - Diagram of lunar horns We draw a khagola circle for same radius (57° 18') as in diagram of eclipse. Directions are marked. Here also, moon is shown as a circle of radius 6 angula (i.e. 12 angula diameter). When moon is in east kapala, sphuta valana is given in eastern point and if moon is in west half of sky, valana is given near west point in its direction (north or south). Valana of sara is given in opposite direction from valana given earlier in both kapala s (east or west half of sky). End point of sara valana is assumed to be sun and from that, a line is drawn upto centre of moon and extended. The point where it cuts the circumference of moon will be the border point between bright and dark portions of moon due to sun. Difference of moon and sun in kala is divided by 900 to get the angula width of bright portion. From centre of moon, on the sun line (the end 720 Siddhanta Darpana point of sara is sun), we give two points on kha- vrtta at 90° distance from sun on both sides. From these points also two lines are drawn to the centre of moon. These two points cut moon on ends of a diameter. On sun line, from circumference, a point at distance of width of bright portion is given. To draw a circle through these points, we draw arcs with 5 angula radius from each of three points. They form two fish figures, whose head - tail lines cut at the centre of circle through these points. From the arc through the three points, the portion towards sun will be the bright portion of moon. In sukla paksa, if moon is in western sky, or in krsna paksa moon in east sky, the bright side will be towards sun point. For sukla paksa moon in east, krsna paksa moon in west, the bright side of moon will be on opposite side of sun point. In sukla paksa, less than half bright moon will be shown by putting the diagram on west side wall. Horns will be bent towards north or south. In krsna paksa, this will be shown on eastern wall. For more than half portion of moon bright, it will show elevation of dark horns. Notes : (1) Valana of moon depends on aksa and ayana valana and due to its sara from ecliptic. Hence both are marked. Since moon circle is of 12 angula diameter, complete diameter 12 angula will be bright when moon - Sun = 180° = 10,800 kala lunar Horns 721 Hence 1 kala difference = 12 10,800 angula bright part 900 bright part. This assumes that moon's speed is constant, which makes little error. But another assumption is that bright part is proportional to angular difference between sun and moon. Actually, it is proportional to utkrama jya i.e. R (1-cos ) as shown below. #■.•■. L 1 ' ■ Figure 3 - Phase of moon C is centre of moon, CO is direction of observer, BEAF is the face of moon, perpendicular to direction of observer. CS is direction of sun and EMF is the face of moon perpendicular to CS direction of sun. Thus the portion of moon between EMF and EBF is the bright portion seen to observer. But the circle EMF is seen obliquely by observer, hence it is seen as half ellipse as projected on BEAF plane whose major axis is EF and semi minor axis is CN. This projected ellipse ENF is the internal boundary of bright portion. CM is radius of moon hence equal to CB and CN is projection of CM, hence 722 Siddhanta Darpana CN = CM cos MCN = CB cos SCC Because iLMCN is the angle between planes which are prependicular to the directions of observer and sun. Hence bright portion NB = CB - CN = CB - CB cos SCC = CB (1 - cos SCC) = CB. Utkramajya SCC Z.SCC is roughly the angle between direc- tions of sun and moon. If they are considered in ecliptic it is difference between longitudes. More accurately, it can be found from triangle OCS whose sides OC, OS and CS are known. (3) In first half of sukla paksa, when sun is setting, moon will be in west half of sky as it is less than 90° ahead of sun. Hence, the diagram will be shown on west wall with direction of sun downwards. In later half of sukla paksa, moon will be in east half of sky, hence its dark horns will be shown in east sky because more than half part in bright. Verses 44-61 - Modern method of showing lunar horns. Thus the method for finding lunar horns has been described according to old siddhanta texts. Now, I describe accurately, ob- served bright part of moon according to my experience and logic. Jya of difference of moon and sun ra& etc is multiplied by yojana karna of moon and divided by yojana karna of sun. Result in kala etc is added to moon of sulda paksa and substracted from moon of krsna paksa. Ihat will be sphuta moon. From this sphuta moon, rasi of sun is again substracted and utkxarna jya is found. That is lunar Horns 723 divided by 573. Result in angula etc. will be measure of bright portion or dark portion which ever is less than half. When less than half of moon is bright, this resultant angula will be marked as bright portion. If more than half is bright, then bright angula measure is (6 angula - the result). As before, from the end point of sara valana in the direction of sun, three points on bright dark boundary are found. Through fish lines, we find ihe centre and draw a circle through these points. Bright portion of moon less than 1-1/2 angula Yt.e. l/8th of moon's diameter) is not seen, because |he end portions of horn are very thin. Increase |n phase of moon, or its decrease should be shown Jo people through diagrams. On 4th day of bright half (sukla paksa), at the sunset time, moon circle is drawn in north direction on earth's surface. 4 diameters are drawn through directions points and angle points. All the diameters bisect each other at the centre. West from moon at a distance, sun is shown. Due to this sun, west half of moon will be bright. To see the bright portion, earth point is given at a distance of 5 hands (5 X 24 angulas) from moon's centre in agni kona (south east direction). North of this earth point will indicate zenith of sky. Though half the moon is always lighted to sun, the portion seen from earth is much ptess than half due to angle between white circle and visible circle. From southeast direction, we see the diameter through nairtya (south west) and isana points (north east). Of the bright portion touching |lP*e north south line, lower half portion will be from earth. ;a - M 724 Siddhdnta Darpana The line from earth centre to south point of moon touches west point and cuts the north east - south west line. From this point in direction of south west, bright portion will be seen. Rest part upto north east point will be dark. When difference of moon and sun is 45°, ancient texts, have assumed 3 angula bright portion. But in this calculation only (1/45) angula is actually seen. Hence, scholars calculate the bright portion of moon from utkrama jya only, because in a sphere, any object will be seen in line of sight (in perpendicular plane only). Notes : (1) Use of utkrama jya - The formula as proved in previous section is through utkrama jya as shown, Logically we can infer it because we see the sphere from curved side, not from side of centre. Hence the distance of plane surface will be proportional to utkrama jya, from centre it is proportional to kotijya. (2) Proof of the formula- As in figure 3, we need to know the angle C C S as seen from moon between directions of sun and direction from observer. L SCC = L CSO + L COS - (1) L COS is the angle be- *' tween directions (rasi) of sun Figure 4 . angle from moon and moon). between sun and earth In sukla paksa (moon - sun) is less than 180°, hence it is smaller angle COS itself. In krsna paksa it (moon - sun) is more than 180°, hence we lunar Horns 725 calculate the outer angle (360° - ZCOS) Then ZSCC = iiCOS - ZCSO - - - (la) Now in ACOS, by sine' rule PC OS OS Sin CSO " Sin OCS ~ Sin SCC as Sin SCC = Sin (180°-OCS) = Sin OCS or, Sin CSO = OC/OS Sin SCC ^ L CSO is very small, because OC is very |mall compared to OS. Hence Sin CSO = L CSO sin L SCC = L SOC approximately. Then L CSO = OC/OS (L SOC) (2) Here, OC = candra karna, OS = surya karna if- Putting the value of L CSO in (1) or (la), we get the SCC whose utkrana jya is to be found. (3) Ahgula value of bright part. For angle of 90°, utkrama jya is 3438 kala and bimba is 6 ahgula bright Hence for utkrama jya 1 kala, 6 1 K: bright portion is (4) Diagram 3438 573 ahgula hi $:■ ■■ ■ ir- tr *■■■ m ToEMti Figure 5 - Bright portion smo from earth 726 Siddhanta Darpana NWS is the face to wards sun and is bright. Face SW-S, E - NE is towards earth. Hence west of point S only, bright portion of moon is seen. WS line is hence the boundary of bright portion, it cuts SW-NE line on K. Hence from K to SW, is bright portion and remaining part from K to NE is dark portion. (5) Modern method - The great circle from sun's centre to moon's centre is perpendicular to line joining lunar horns. The great circle from zenith to centre of moon is at angle from sun moon great circle, which is the angle of lunar horns with horizon. This angle can be known from spherical trigonometry, as discussed in triprasna- dhikara. Figure 6 (a) Figure 6 (b) IN figure 6 a, NZS = yamyottara, Z = Zenith (Khasvastika) O = observer, NOS = north south line NWS = western horizon M = Moon in western sky R = position of setting sun ZM = natamsa of moon MR = Distance between sun and moon lunar Horns 727 L RZM = Difference between directions of moon and sun (digamsa) from zenith. Natakala can be known from visuvamsa (rising times) of sun and moon and their kranti. cos (nata kala) cos (natamsa) - Sin (akSamsa) x Sin (kranti) cos (aksamsa) x cos (kranti) This equation will give the natamsa Then digamsa will be known from the following equation cos (digamsa) cos (dhruvantara) - cos (natamsa) x Sin (akSamsa) Sin (natamsa) x cos (aksam§a) These equation have been derived for calcula- tion of natamsa and calculation of karna vrttagra in Triprasnadhikara verses 71 notes (3) and verse 44 (notes). Thus in spherical triangle ZMR, we know ZM, ZR, MR, and L MZR. ZR = 90°. Hence we can know L ZMR and the elevation of lunar horns. In figure 6(b), M = centre of moon OM = vertical circle of moon centre (drk=mandala) RM = Direciton of sun from moon ANKBC = Bright portion of moon L OMR -L AMN = angle of elevation of lunar horns In figure 6 (a), from spherical trigometry cos MR = cos ZR cos ZM + Sin ZR Sin ZM Cos L RZM After finding MR from this equation. 728 Siddhanta Darpana Cos ZR - cos ZM x cos MR cos L ZMR = SinZMxSinMR 180° - L ZMR is the angle of elevation of horns, because it is equal to L PMR. If sun is north from moon, then north horn will be upper and if south, then south horn will be upper. If digamsa of both sun and moon are same then horns will level. After knowing this, diagram of horns should be drawn as per figure 6(b). Verses 62-63 : Horns of budha and sukra also are visible through telescope. In India, north horn is mostly seen higher in both west and east kapala. Very rarely, south horn is seen higher. Notes : For aksamsa more than 28-1/2° north, both sun and moon will be always in south. As we see from north, northern portion of bright horn will look bigger. Verses 64-67 : Reasons for new methods — Earlier astronomers used to find difference of kranti's of sun and moon through a sanku of 12 angula and from that, elevation of horns was found. Since this method doesn't give results as observed, I am rejecting it. When sun is prepen- dicular to equator, half disc of moon in sayana makara beginning is seen cut by meridian line at zenith. Hence, half disc is seen bright. Hence utkrama jya of (moon-sun) is multiplied by 1st sphuta gati of moon and divided by (173452). Result in kala is added or substracted from half disc of moon to find the bright portion width. This is added to half diameter when bright portion is more than half, otherwise substracted from it. This &*":.■ 729 lunar Horns w fll be correct measure of bimba in both east and west sky. When moon is at 11° from sun, its light is more than budha bimba of diameter 17 vikala and less than bright bimba of guru. Hence, it is not proper to consider heliacal rising and setting of moon at 11° kalamsa difference. The author considers it to be between 11 P and 12°. At the end of 1st day of bright half, 108th part of moon's disc is seen, even though it is very thin. On 4th day, its 1/6 part will be seen bright. hi the end of 5th day, 1/4 parts will be bright. At the end of tenth day 3/4 part will be bright. On purnima, complete disc will be bright. On 8th day end* half disc and on 11th day end 5/6 parts will I appear bright. ^ Notes : (1) Brightness of moon has been calculated according to value of utkramajya for the angle between sun and moon. (2) Madhya bimba kala x madhya karna of candra = spasta bimba kala x spasta karna Hence bright portion in kala spasta bimba kala x utkramajya 2 X trijya (= madhya karna) madhya bimba kala x utkrama jya * 2 X spasta karna = madhya bimba kala X utkramajya sphuta gati 2 X trijya X madhyagati &' fe b ■ >■; ■ 730 Siddhdhta Darpana - utkramajya X sphuta gati madhya bimba kala 2 X trijya X madhya gati = utkramajya X sphuta gati 444 x trijya 48705 X X 2 x trijya x 790/35 utkramajya x sphuta gati = ~~ 173452 "as given Verses 68-69 - Prayer and end On sea beach, Lord Jagannatha protects people from anger of yama with his sudarsana cakra, and destroys aU diseases borne out of desires. May he end all our illnesses due to passions. Thus ends the fourteenth chapter describing elevation of lunar horns in Siddhanta Darpana, written for consonance in calculation and observa- tion and education of students, by Sri Candrasekhara born in famous royal family of Orissa. £v Chapter - 15 MAHAPATA VARNANA Verse 1 - Scope - I am describing mahapata !# told in scriptures, which destroys the good deeds (karma) earned in pilgrimage, sacred thread wear- marriage etc, in whose discussion mathe- Hans are also confused, and on whose C]- occasion, results 01 cnaruy, jap* anu uaui i/™^«- auspicious as in an eclipse. Notes : Mahapata is a fictious conjunction of Sun and moon and is as good or bad as an eclipse, lestroys results of good deeds which accrue due marriage etc as described in scriptures. But if |§pod works like charity are done during mahapata, pfhey are as fruitful as in eclipse. This is a difficult lopfc, as the conjunction is observed only mathe- matically not as a real phenomenon. Verses 2-8 : Two mahapatas - Patas are of two types - Vaidhrti and vyati- pata When their (sun moon) krantis are equal, then these patas occur. Out of gola and ayana, if ayana ie, then pata is vaidhrti and if gola is same then it is vyatipata. When moon and sun are in same diurnal they have gola sandhi. When both are in ce of parama kranti, they have ayana sandhi. When moon and sun are in one gola but different ayanas and their krantis are equal then it is vyatipata yoga. ■'.'<-■ 732 Siddhdnta Darpana When moon and sun are in different gola but same ayana and their kranti are equal, it is vaidhrti yoga. When kranti is same, their aspects are added (i.e. they are at same angle with equator plane). (In Surya siddhanta) when moon and sun both have same kranti, due to combination of their rays at same angles there is flow of fire which is destructive for living beings. Atipata yoga is always bad and destructive. Other names of this yoga are vyatipata and vaidhrti. Each pata has dark colour, very ferocious body and red eyes. Both are valiant and occur every month. Pata from spasta position (of moon and sun) is more destructive than pata from mean position. (Quotation ends) Notes (1) Two yogas are named vaidhrti and vyatipata, but these have no relation at present with the two mahapatas. However, these can be calculated from sum of longitudes of sun and moon and in that way they are related to yoga cycle. dinemal circle of sun diurnal circle Equator diurnal circle of moon Figure 1 - (a) Vyatipata Figure 1 - (b) Vaidhrti Vyatipata is 10th yoga and vaidhrti is the last. tfghapata Varnana 733 Figure 1 - Mahapata Figure 1 (a) shows vyatipata when, moon and sun have common diurnal circle i.e. same kranti tyxt at the other end of orbit. Figure 1 (b) shows, vaidhrti yoga, in which the kranti of moon and sun are equal and opposite, |.e. diurnal circles of moon sun are at equal distances from equator, but in opposite direction. Sun and moon are in same side of sphere i.e. in same gola. In these figures Y is 0° sayana mesa and Q Us pata of moon i.e. rahu. £ is inclination between §|quator and ecliptic and § between ecliptic and Inoon's orbit. If latitude of moon's orbit is neglected (it is |ess than 5° always), both moon and sun are on eelitpic. If their longitudes S 2 of sun and M 2 for pinoon from sayana 0° are taken, then for equality I ©f kranti. I Sin Sl = Sin Ml I = Sin (180° -M L ) I or S L = 180° -Ml or S L +M L = 180° ?■■"■ I When they are numerically equal but in | opposite direction, then Sin S L = - Sin M L = Sin |<360°-Ml) I . Hence S L + M L = 360°. (2) In vaidhrti yoga, Sl + Ml - 2 ayanamsa = 360 180 For Vyatipata yoga Sl + Ml - 2 ayanamsa = Thus vaidhrti yoga coincided with vaidhrti f ***ahapata when ayanamsa was 0°. But vyatipata 734 Siddhdnta Darpana yoga is only 10th yoga starting at 120° and it will not tally with the mahapata. Verses 9-15 : Calculation of yoga - When sum of sayana sun and sayana moon is 12 rasi then vaidhrti yoga is near. Similarly, if sum of sayana sun and sayana moon is 6 rasi or 18 rasi, then vyatipata yoga is imminent. When sun and moon are in different quad- rants of ecliptic, then only pata can happen. Both vyatipata and vaidhrti yogas occur once each ( month. In some months vyatipata occurs twice, sometimes it doesn't occur in a month. Pata are possible when viskambhaka etc yogas occur. For that, we multiply ayanamsa by 2 and added to minutes (kala) of a circle (21,600) or half circle (180° = 10,800') if substracted earlier, and substracted if added earlier. When result is more than (21,600), cakra (21,600) is substracted/ By dividing it with (800), result will be past no. of yogas from viskumbha etc. Adding 1 to quotient it will give the number of current yoga. Remainder multiplied by 60 and divided by 800 gives the part of current yoga lapsed. If moon has no sara, then it is also the time of pata. When moon has sara, pata will be slightly before or after this time. Hence we should roughly calculate pata, first according to madhyama kranti (i.e. Kranti of ecliptic point of moon without sara). Notes : If has been explained earlier that patas will occur when sum of sayana moon and sun is 6 rasi (for vyatipata) or 12 rasi for vaidhrti. Mahapata Varnana 735 For, if Sl amd Ml are sayana longitudes of $un and moon, when their krantis are equal, for vyaupata Sin S L = Sin M L = Sin (180°-M L ) or S L = 180°-M L or S l + M L = 180° (1) When kranti is equal and opposite for vaidhrti Sin S L = - Sin M L = Sin (360°-M L ) or Sl+Ml = 360° (2) While pata is calculated with sayana sun and moon, assuming madhyama kranti without sara, yoga is calculated for nirayana moon and sun. Hence, if ayanamsa is A and nirayana moon :and sun are S and M, then >-'■' (S + M) Kala y°8 a= SOOKala " * " ' (3) because each yoga extends for 800 kala of sum of sun and moon position. S L = A+S, Ml = A+M Hence for patas S L + M L = 6 rasi or 12 rasi (180° or 360°) or (S+A) + (M+A) = 6 or 12 rasi or S+M = (6, or 12 rasi) - 2 A Putting this value of S+M in kala in (3), we get the yoga number as stated in text. Formulas (1) and (2) give kranti depending only on ecliptic. Since sara is very small it will be approximate time of pata also. As ayanamsa remains almost constant, the yogas for occurrance of pata are fixed for some years. We can thus know the approximate time of pata by the current yoga. After knowing sthula pata, we get it corrected 7 36 SiddMnta Darpana for sara of moon to know when sphuta kranti is equal. (2) Since yoga is sum of sun and moon, it changes with sum of speeds i.e. (790/35+59/8) = 849/43 average speed. At this rate, rotation takes 21,600 Kala = ^ approximately 849/43 kala/day In a lunar month of 29.5 days it will definitely complete one cycle, hence both the patas will occur once at least. Due to extra length of lunar month, sometimes, one pata may occur twice. If true kranti of moon is more than 23-1/2°, a pata may not occur. Verses 16-20 : Sthula pata for present ayanamsa. At present (1869 AD - writing of book), ayanamsa is 22°. (It can be almost same in 1996 also with only 1° difference). Hence in sukla (24th yoga) and vrddhi (11th yoga), vaidhrti 3rd quarter and vyatipata 1st quarter often fall. That is their madhyama time. Hence, we assume the sukla yoga and vrddhi yoga as cakra (21,600) and cakrarddha (10,800 kala) aproximately. On the day of that yoga, we calculate accurate value of sun and moon (at the end ot these yoga times). Ayanamsa is added to both sun and moon. Then we find the difference of (sayana sun+moon) from (10,800) or (21,600) Kala. That wiU be divided by sum of' sun and moon gatis and multiplied by 60 to get time in danda etc. (It can be calculated from proportionate duration of cur- rent yoga also) This time is added to time of sukla or vrddhi yoga if (sun+moon) was less than that, otherwise it will be substracted. Mahapata Varnana 737 After successive approximations, sum of say ana sun and say ana moon will be equal to 6 or 12 rasis at the calculated time. Then we calculate the sara of moon. These yogas are not visible, hence drkkarma or lambana, nati are not needed for moon. Pata is calculated from earth's centre only. Notes : According to method described after verse 15, the yogas at the times of pata have been calculated (based on madhya kranti of moon, assuming zero sara), for 22° ayanamsa. At present also for 23-1/2° ayanamsa, it is almost same. Accurate time of madhyama pata is found by method of successive approximation. Verses 21-33 - Pata from sphuta kranti (From Surya siddhanta) - In odd quadrants, if sphuta kranti of moon (i.e. kranti of its ecliptic point corrected for sara) is more than kranti of sun, then pata has already passed. If sphuta kranti is less, then pata is yet to come. In even quadrants if sphuta kranti of moon is more than kranti of sun, then pata is to come, if it is less then pata has passed. Persons conversant with gola (spherical trigonometry) can know the time of sphuta pata through their methods. But detailed calculation method is explained for common men. When sum of rasis of sun and moon (both Myana) is exactly squal to cakra or cakrardha kala ^,600 or 10,800 minutes), then if pata has lapsed, 60 danda is substracted from that (mean pata) me. If pata is yet to occur, then 60 danda is in*" 7 3£ Siddhanta Darpana added to that time. For that revised time, we calculate sun, moon's pata and sara and difference between sphuta kranti of sun and moon. If sign of (candra kranti - sun kranti) has changed after this revised time then, pata has occured during this 60 danda interval. If sign is same, then pata is beyond that interval. To find the correct time of pata, we find the difference of krantis of sun and moon, both at the mean pata time and at interval fo 60 danda. If they are of different sign, they are added. If difference is of same sign their difference is taken. This will be the first kranti gati for finding pata. First kranti difference in kala is multiplied by 60 and divided by first kranti gati. Quotient in danda etc is added to mean pata time, if pata was to come and substracted from it, if pata had already passed. At first corrected pata time, we again find the kranti difference of sun and moon and find the second kranti gati kala. Kranti difference of 1st corrected pata time is multiplied by the time difference and divided by second kranti gati. By the result in ghati etc, we again correct the 1st corected pata time. Kranti gati is found by multiplying the change in kranti difference by 60 and dividing by the time difference. By repeating this process, by successive approximation we get the time of mid-pata. Last kranti gati wil be the gati of kranti antara at mid pata time. Mahapata Varnana 739 B V + C Figure 2 - Mahapata Notes : (1) Whether pata has gone or not - In figure 2, VBAC is the ecliptic where V is vernal equinox, (or say ana mesa 0°) and A is autumnal equinox. B and C are solstice points in summer and winter at 90° from these. Thus the 1, 2, 3, 4th quadrants from V are VB, BA, AC and CV in the direction of motion shown by arrows. When moon is in odd quadrant (VB or AC), eg. M in VB, then for vyatipata sun is at Si so that VM = ASi and VM+VSi = VM + VA - AS = VA = 180°. Similarly for vaidhrti, sun will be at S2 in VC where VS2 = VM. Thus sun will be in 2nd or 4th quadrants i.e. in even quadrants. At V and A, kranti (madhya kranti for moon) is zero, in VB portion it increases in north direction and in AC portion in south direction. Thus the kranti increases in VB and AC which are odd quadrants and decreases in the even quadrants BA I and CV. Thus when moon is in odd quadrant and its true kranti is more than sun (when madhya kranti is equal) then kranti of moon will further increase and sun will decrease for even quadrant. Hence they will be equal at an earlier time i.e. pata had &■ 'ft/ 740 Siddhanta Darpana already passed. If moon's true kranti is less than sun, it will increase and sun's kranti will decrease and they will be equal after some time. Hence pata will come after some time. When moon is in even quadrant, sun will be in odd, hence moon kranti will be decreasing and sun kranti will be increasing. If moon's kranti is more, it will be equal to sun after some time and spasta pata will come. If moon's kranti is less, pata has already passed. This analysis has considered increase of only mean kranti of ecliptic point. Sara of moon also changes, which will change the true kranti. Hence, for correct calculation, moon's sara also has to be calculated. Suppose moon in first quadrant has 5° north sara (maximum value). Then its true kranti at madhyama pata will be 5° more than sun. Moon's pata decreases at average rate of 5/6.8 degrees perday, because its quarter revolution is in 27.3/4 = 6.8 days. Kranti of sun will decrease and madhya kranti of moon will increase at the rate of 23.5/91 = 1/4° per day approximately. Hence total increase in moon's Kranti will be 5/6.8 + 1/4 + 1/4 per day = 1.24° per day compared to sun. Thus the true pata will be about 4 days before madhyama pata. Suppose it is vaidhrti pata. If previous vaidhrti is 3 days later, then they will be in 25-7 = 18 days and in 12 days of the month another pata can occur. Thus there will be two vyatipata which comes about 12 days after vaidhrti and one has already passed between two vaidhrtis. Mahapata Varnana (2) Calculation of true pata time 742 To + 60 Fig - 3a To + 60 Fig - 3 (b) B Figure 3 - Calculation of true pata time To is the madhyama pata time when mad- hyama kranti of sun and moon are equal. But the true kranti of sun and moon are unequal due to sara of moon. Let AT be the difference of kranti of moon and sun at time To (it is negative if kranti of moon is less). To make a first aproximation of true kranti time, we calculate the position at 60 ghati difference according to the pata time is earlier or later. Then kranti difference is BB' where B' indicates time To ± 60 danda. When kranti difference has same direction i.e. at O (To) and B' (moon-sun kranti) is both positive or negative, the true kranti will be equal at time C outside B' i.e. outside the interval of T to To ± 60. We assume here that kranti difference has same gati (or rate Of change), hence it will be zero where line AB Guts the line OB' where kranti diff. is zero. This is shown in figure 3(a). Figure 3(b) shows that the sign of kranti diff. ;es. Then AB line cuts OB' between the Interval at C. 742 Siddhanta Darpana In both the figures we draw a line BA' parallel to OB' which cuts AO (or AO extended in fig b) at A' Then AA' is change in Kranti diff. in 60 danda time. Here AA' = AO-BB' = diff of kranti diff in fig (a) when kranti difference has same sign. AA' = AO + OA' = AO .+ BB' = sum of kranti diff. in figure (b) when they are of different signs. Thus speed of kranti diff. is AA760 in each danda. Hence it will be zero in time T Ti * ■ AO x 60 J , = danda AA' Here AA' is the gati of kranti antara in 1 day or 1st kranti gati. Thus we correct the madhya pata time according to difference of kranti antara in 1 day. By calculating the kranti difference again at point Ti we get more accurate value of true pata. (3) Surya siddhanta has given another method, using difference of moon from rahu. Here we have not described the method of calculating sara of moon, which is necessary for sphuta kranti. Sara depends upon bhuja jya of difference between moon and rahu, hence, we take this as difference of sphuta kranti in surya siddhanta. Verses 34-42 - Sparsa and moksa of pata. This was the time, centres of moon and sun were having same kranti i.e. mid point of pata. When the first points of moon and sun have equal kranti, this is called sparsa time as in eclipse and When the last point has equal kranti it is moksa time. Thus full pata time is from sparsa to moksa. Mahapata Varnana 743 Now, method to find sparsa and moksa time is being described. Like method of lunar eclipse, we find the bimba of moon and sun at mid time of true pata and add their semi diameters (manaikyardha),. Sum of semi-diameters is multiplied by 60 and divided by last kranti gati (i.e. kranti difference gati at mid pata time). Result will be madhyama sthiti ardha time in danda. By adding to pata mid time, we get moksa time and by substracting we get the sparsa time. At these approximate times of sparsa (or moksa), we again find the difference between sphuta krantis. If this is less then manaikyardha (sum of semidiameters of sun and moon) then sparsa time has passed (or moksa time is to come), as the kranti difference decreases from sparsa time (equal to manaikyardha) to mid time of pata, where it is zero. When kranti antara is more than manaikyardha, then sparSa is to come (or moksa time has passed). Kranti antara at sparsa (or moksa) time multiplied by 60 and divided by madhya sthiti ardha will give 1st kranti antara gati at sparsa (or moksa) Manaikyardha vikala at sparsa (or moksa) is divided by first kranti antara gati at sparsa. It will give sthiti ardha for sparsa (or moksa) in danda. By substracting them (or adding) to mid pata time, we get the time of sparsa (or moksa) - 1st sphuta value. 744 Siddhdnta Darpana Now at the first sphuta value of sparsa (or moksa), kranti antara kala is multiplied by second sthiti ardha time in ghati for sparsa (or moksa). We get second kranti antara gati at sparsa (or moksa). Again we can get second sphuta value of sparsa (or moksa) times and s'thitiardha. By successive approximation, we get the steady value of sthiti ardha etc. Moksa Figure 4 - Sparsa and moksa of pata Notes : In figure 4, fictitious joining of sun and moon have been shown. Kranti' s are equal in a pata, but they may be in different direction. They have been shown in same direction. Sun and moon are always in adjacent quadrants for pata as shown in figure 2, but they are shown at one place for explaining equality of kranti of different parts of sun and moon. At point O, kranti of centres of sun and moon are equal. Hence, it is mid time of pata. When centres of moon and sun are at Mi and Si, their kranti antara is Mi Si and discs touch each other at Ti. Kranti difference is moving along Mi OM2. At position M2, S2, the discs just touch at T 2 . Thus before Mi or after M2 position, the krantis of no part of sun and moon will be same and there will be no pata. Between these positions, Mahdpata Varnana 745 some point of moon will have same kranti as some point of sun, hence it will be pata. It is clear that at sparsa time Kranti antara M1S1 = M1T1 + Ti Si = semi diameters of (sun + moon). Similarly at moksa time also it will be equal to manaikyardha. First we calculate the approximate position Si, S 2 from the kranti antara gati at O. Then we correct it with the gati at approximate positions of Si and S 2 to get more correct value. By repeating this process, we get the accurate value. This is only a diagram to explain equality of kranti of different points, there is no closeness of sun or moon as in ecliptic. Verses 43-45 : Effects of pata - According to Surya siddhanta the time of pata from beginning to end is fiery like burning fire and all auspicious works like marriages, sacred thread ceremony etc are prohibited during this. Pata arises due to equality of kranti' s of sun and moon. That destroys all results of noble deeds. By knowing the period of pata, penances like bath, charity, mantra, sraddha, worship, offers in fire all give good results, as on the occasion of eclipse. Verses 46-49 : Duration of pata Average duration of pata is two prahara or 15 danda. Minimum duration is 9 danda and • • • • maximum duration is 2/20 days. 74^ Siddhanta Darpana When ucca and pata of moon is in the last part of 12th rasi (mina), sayana sun is near mina and sayana moon is near kanya, then pata duration is minimum. When candra, its ucca, and pata all are at beginning of karka rasi, sayana sun is at the end of dhanu rasi, then pata is for maximum duration. If pata of candra is between mithuna 28° to karka 1 ° instead of karka beginning, then there is minimum time between two equalities of krantis (i.e. mahapatas). If at the end of ayana, spasta kranti of moon is more than 28°, then krantis cannot be equal. Notes : (1) Here pata has been used as short form of two mahapatas - Vaidhrti and vyatipata - when kranti of sun and moon are equal. But pata means the point of inter section of a planets orbit with ecliptic which is sun's orbit. For moon's orbit, pata's are called rahu or katu. As in case of all orbits, the pata point after which planet starts having north sara, the ascending node (rahu) is pata of moon. When moon, its ucca and its pata are in beginning of karka i.e. 90°, then sun in 270° will cause vaidhrti. Then moon has almost zero sara and its true north kranti is equal to sun kranti in opposite south direction. Speed of krantis will be slowest and speed of moon also will be slowest near its ucca, hence its pata will be longest. If moon pata is between mithuna 28° to karka 1° (i.e. 88° to 91°) then within this movement of 3° pata, just before pata position moon kranti will Mahapdta Varnana 747 be less then 23-1/2° equal to kranti of sun before 270°. After pata, sara of moon will rapidly increase and spasta kranti will be equal to sun kranti maximum at 270°. Hence next pata will come earliest. Opposite to the longest pata, moon at 180° and sun at 360° (vyati pata), if ucca and pata of moon are near 360/ then speed of moon is maximum, 0° kranti period will be for lowest period as kranti speed is maximum at 0° kranti and sara. hence pata is of smallest duration. For such situations, maximum and minimum periods of patas have already been given. (2) Maximum kranti of sun can be only upto 23-1/2° in either direction. However, due to sara, moon can have kranti upto 28-1/2° due to its parama sara of 5°, when madhya kranti and sara both are maximum and in same direction. Then moon's kranti will be between 23-1/2 to 28-1/2° and sun's kranti will be always less than 23-1/2°. Hence true krantis can not be equal and there can be ng true pata, though madhya pata will occur. Verses 50-54 : Gola and ayana for pata For calculating true pata, sara of moon changes due to its orbit (distance from its pata rahu). But madhya kranti is same as kranti of sun in that ayana. At gola sandhi (zero sara) sphuta kranti doesn't change due to sara. But in ayana sandhi (maximum kranti but least kranti speed), kranti gati changes due to sara gati. Reason is that kranti gati is more in gola sandhi (at equator) and least in ayana sandhi (maximum kranti position). 748 Siddhanta Darpana • In south and north gola, north south motion of moon due to sara doesn't change its total kranti gati. Being deflected north or south due to pata, moon still continues its motion on kranti vrtta. It is not affected, whatever may be the value of sara. Varahamihira has described gola and ayana system for mahapatas very logically in his Brhatyatra book. Notes : This is an objective description and needs no further comment. Brhatyatra is not a well known book of Varahamihira who has written three texts in three branches of jyotisa - Brhatsarhhita (Sarhhita), Brhatjataka (astrology - phalita) and Panca siddhantika (astronomy). Verses 55-58 : Inauspicious times In grahasphuta - chapter 5, 27 yogas have been described according to sum of rasi etc of sun and moon. Out of them 27th yoga is vaidhrti and 17th is vyati pata. These yogas are very fiery because sun and moon become very angry, their aspects being inclined at same angle to equator, in same manner as two bullocks become angry when they are forced to move together round a pole for crushing oil seeds or separating grain chaff. From Surya siddhanta - Last quarters of aslesa, jyestha and revati - rasi and naksatra both have their borders. Hence last quarters (l/4th part) of these naksatras is called ganda. Half of first quarter (first l/8th part) of^next naksatras (magna, mula and asvinl) are called gandanta. All auspicious works are prohibited in sandhi (junction) of rasis. Last navamsa of karka, vrscika Mahapata Varnana 749 and mlna rasi are in mlna rasi. First navamsa of next rasis (sirhha, dhanu and mesa) falls in mesa rasi. Hence all these navamsa are also bad. Like gandanta, these navamsa also fall in the junction of rasi and naksatra, hence good works are prohibited in them. Visti (bhadra) etc bad karanas are also to be avoided. Notes : This has nothing to do with ganita jyotisa. This can be considered use of these calculations of pata, naksatra karana and yoga. Surya siddhanta explains that 3 vyati patas, 3 rasi sandhi and 3 naksatra sandhi all are very bad. Here 3 types of vyatipata are - mahapata called vyatipata and vaidhrti, yogas named vyati pata and vaidhrti. Mahapata are of two types - one from mean value of kranti and one from true kranti, hence three types of vyatipatas. 12 rasis or 27 naksatras both are equal to 360° or full circle. Hence 1 rasi is equal to 2-1/4 = 9/4 naksatras. Thus when 4 rasis are complete, 9 naksatras also are completed, and their junctions combine. To tally rasi with naksatra, each naksatra is divided into 4 quarters, so that each rasi has 9 quarters. Each rasi is also divided into 9 parts called navamsa. Thus, 1 navamsa = 1 quarter naksatra = 3° 20'. Navamsa also is counted like rasi starting with 1st navamsa of mesa as mesa, 2nd navamsa as vrsa etc. • » • Thus at the and of 4, 8, 12 rasis, 9th, 18th and 27th naksatras i.e. mlna navamsa is completed. Next navamsa i.e. 1st navamsa of 5, 9 1st rasis are 750 Siddhdnta Darpana mesa navamsa. According to rules stated, last quarter of 9th, 18th and 27th rasis or first half quarters of next naksatras are bad. If a child is born during this period (i.e. if moon is in ganda or gandanta naksatra), that naksatra is worshipped when it comes again (on 27th day of birth). As the seventh day Sunday was not meant for work in Christianity, 7th karana visti is not good for starting any important work or for proceeding on a journey. It is also called bhadra (meaning good - probably for holiday purpose). Verses 59-62 : Comments on the siddhanta methods - Brahma took 47,400 divine years in creation of world, which is called srsti kala (creation period). From next day after creation, revolutions of graha, their ucca and pata etc started. Hence it has aheady been stated that for calculation of graha etc, the years of creation will be deducted from the years counted from beginning of kalpa. After completion of creation, caitra sukla pratipada was the first tithi. Then sun was rising in Yamakotipattana and it was mid night in Lanka. This day was named as ravivara (sunday). From that instant Brahma left graha, ucca and pata to move in their orbits from first point of asvini naksatra (mesa 0°) From that time only days, months, years, kranti and revolutions of graha etc started. They had not started from start of day of Brahma (called kalpa). From that time, only ghati .(1/60 of a civil day), yuga and manu etc started. Sages like Parasara have described king, ministers and protectors of the years, clouds like drona and puskara, rulers of grains etc, parts of Mahapata Varnana 751 fire, rain and deceases, raja yoga etc for predicting good or bad results of future. Sometimes, they give the said results, sometimes they don't. Due to that, these have not been described here, as in other siddhanta texts. Sun and moon complete their revolutions at the end of every yuga and also in l/4th part of yuga. During a quarter of yuga (10,80,000 years), savana ahargana (civil days) are (39,44,79,457). At the end of dvapara, srstyabda (years since creation end) was (1,95,58,80,000). This divided by years of a quarter yuga (10,80,000) gives quotient (1811) and zero remainder, hence there is no need to state dhruva (positions) of sun and moon at the end of dvapara (after complete revolutions they are again at start of mesa 0°). Verses 63-66 : Start of Karana for this book From beginning of creation to dvapara end, past years (years completed at entry of mean sun in mesa) were (1, 95, 58, 80, 000), and at (4970) completed years in kali (1869 AD - Karanabda) the aharganas are (7, 14, 40, 22, 96, 627) arid (18, 15, 334) from creation and kali. Both are correct as checked by vara (weekdays). At beginning of karanabda, when mean sun had entered mesa, first day according to mean value (sun and moon) was caitra sukla pratipada. The dhruva stated for that day (mean positions at beginning of year), when added to daily motion for lapsed days, becomes dhruva of madhyama graha acording to surya siddhanta. Ahargana of karanabda starts with tuesday (mangala vara). 752 Siddhanta Darpana The day before beginning of karanabda has been assumed monday. That day was caitra sukla pratipada. (mean speed). According to spasta position it was vaisakha adhimasa (extra month) pratipada. Hence the day before start of karanabda is correct caitra pratipada according to mean speed and monday, which is convenient day for stating dhruvas. From starting point of karanabda, (18, 15, 334-15) days before, kaliyuga had started at mid night at Lanka. According to ancient authorities, that was caitra sukla pratipada by mean positions. Again first day of karanabda is in vaisakha by true position. To find this caitra sukla pratipada, dhruva at t